A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 3, (3 n + 2) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 3 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 3 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(3 n + 2) Define , 3, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 3 n if n modulo, 2, is , 0, then Q(n) equals , --- + 1 2 3 n if n modulo, 2, is , 1, then Q(n) equals , --- + 3/2 2 d(n) modulo, 3, can be computed VERY fast (in logarithmic time!) using the fo\ llowing recurrence, taken modulo, 3 d(3 n + a) = Q(3 n + a) + C[a] d(n) subject to the initial condition d(0) = 1, d(1) = 1, d(2) = 2 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 3, is 2, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 2, 0, 0, 2, 1, 1, 1, 2, 0, 2, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 2, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 2, 0, 0, 1, 0, 1 ----------------------------------------------------------------------------\ ----------------- This took, 0.041, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 4 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 4, (4 n + 3) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 4 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 4 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(4 n + 3) Define , 5, integers as follows: C[0] = 0, C[1] = 1, C[2] = 3, C[3] = 3, C[4] = 1 If j is larger then, 4, then C[j]=0 Also let Q(n) be the polynomial, 2 n + 2 d(n) modulo, 4, can be computed VERY fast (in logarithmic time!) using the fo\ llowing recurrence, taken modulo, 4 d(4 n + a) = Q(4 n + a) + C[a] d(n) + C[a + 4] d(n - 1) subject to the initial condition d(0) = 2, d(1) = 2, d(2) = 0, d(3) = 2, d(4) = 0 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 4, is 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0 ----------------------------------------------------------------------------\ ----------------- This took, 0.018, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 5 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 5, (5 n + 4) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 5 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 5 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(5 n + 4) Define , 7, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1, C[3] = 3, C[4] = 3, C[5] = 1, C[6] = 1 If j is larger then, 6, then C[j]=0 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 5 n if n modulo, 2, is , 0, then Q(n) equals , --- + 2 2 5 n if n modulo, 2, is , 1, then Q(n) equals , --- + 5/2 2 d(n) modulo, 5, can be computed VERY fast (in logarithmic time!) using the fo\ llowing recurrence, taken modulo, 5 d(5 n + a) = Q(5 n + a) + C[a] d(n) + C[a + 5] d(n - 1) subject to the initial condition d(0) = 2, d(1) = 2, d(2) = 4, d(3) = 1, d(4) = 3, d(5) = 2, d(6) = 1 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 5, is 0, 1, 0, 4, 1, 3, 0, 0, 2, 0, 2, 1, 3, 3, 0, 1, 0, 2, 3, 1, 4, 1, 1, 2, 4, 4, 0, 4, 4, 2, 1, 3, 1, 2, 4, 4, 4, 3, 1, 4, 0, 4, 3, 3, 0, 1, 2, 4, 4, 2, 1, 4, 2, 0, 2, 0, 3, 1, 0, 3, 3, 1, 2, 0, 2, 0, 1, 4, 4, 2, 1, 0, 3, 3, 0, 1, 1, 3, 1, 4, 0, 1, 0, 4, 1, 3, 3, 3, 1, 4, 0, 4, 3, 3, 0, 1, 2, 4, 4, 2, 1 ----------------------------------------------------------------------------\ ----------------- This took, 0.021, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 7 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 7, (7 n + 6) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 7 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 7 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(7 n + 6) Define , 11, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1, C[3] = 3, C[4] = 3, C[5] = 6, C[6] = 6, C[7] = 3, C[8] = 3, C[9] = 1, C[10] = 1 If j is larger then, 10, then C[j]=0 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 7 n if n modulo, 2, is , 0, then Q(n) equals , --- + 3 2 7 n if n modulo, 2, is , 1, then Q(n) equals , --- + 7/2 2 d(n) modulo, 7, can be computed VERY fast (in logarithmic time!) using the fo\ llowing recurrence, taken modulo, 7 d(7 n + a) = Q(7 n + a) + C[a] d(n) + C[a + 7] d(n - 1) subject to the initial condition d(0) = 3, d(1) = 3, d(2) = 6, d(3) = 2, d(4) = 5, d(5) = 4, d(6) = 0, d(7) = 2, d(8) = 1, d(9) = 6, d(10) = 1 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 7, is 0, 4, 1, 5, 6, 3, 1, 1, 3, 6, 5, 4, 0, 2, 5, 3, 6, 0, 3, 0, 3, 3, 6, 2, 5, 4, 0, 2, 5, 3, 6, 0, 3, 0, 3, 5, 1, 1, 4, 2, 5, 1, 6, 0, 0, 6, 1, 5, 2, 1, 0, 1, 2, 5, 1, 6, 1, 1, 2, 4, 4, 1, 0, 5, 3, 2, 6, 6, 2, 3, 4, 6, 5, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 2, 2, 5, 0, 3, 0, 3, 6, 2, 4, 0, 1, 4, 4, 5, 4, 3 ----------------------------------------------------------------------------\ ----------------- This took, 0.024, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 8 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 8, (8 n + 7) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 8 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 8 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(8 n + 7) Define , 13, integers as follows: C[0] = 0, C[1] = 1, C[2] = 3, C[3] = 7, C[4] = 5, C[5] = 6, C[6] = 2, C[7] = 2, C[8] = 6, C[9] = 5, C[10] = 7, C[11] = 3, C[12] = 1 If j is larger then, 12, then C[j]=0 Also let Q(n) be the polynomial, 4 n + 4 d(n) modulo, 8, can be computed VERY fast (in logarithmic time!) using the fo\ llowing recurrence, taken modulo, 8 d(8 n + a) = Q(8 n + a) + C[a] d(n) + C[a + 8] d(n - 1) subject to the initial condition d(0) = 4, d(1) = 4, d(2) = 0, d(3) = 4, d(4) = 0, d(5) = 0, d(6) = 4, d(7) = 0, d(8) = 4, d(9) = 0, d(10) = 4, d(11) = 0, d(12) = 4 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 8, is 4, 4, 0, 4, 0, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 0 ----------------------------------------------------------------------------\ ----------------- This took, 0.016, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 9 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 9, (9 n + 8) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 9 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 9 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(9 n + 8) Define , 15, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1, C[3] = 3, C[4] = 3, C[5] = 6, C[6] = 6, C[7] = 1, C[8] = 1, C[9] = 6, C[10] = 6, C[11] = 3, C[12] = 3, C[13] = 1, C[14] = 1 If j is larger then, 14, then C[j]=0 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 9 n if n modulo, 2, is , 0, then Q(n) equals , --- + 4 2 9 n if n modulo, 2, is , 1, then Q(n) equals , --- + 9/2 2 d(n) modulo, 9, can be computed VERY fast (in logarithmic time!) using the fo\ llowing recurrence, taken modulo, 9 d(9 n + a) = Q(9 n + a) + C[a] d(n) + C[a + 9] d(n - 1) subject to the initial condition d(0) = 4, d(1) = 4, d(2) = 8, d(3) = 3, d(4) = 7, d(5) = 6, d(6) = 1, d(7) = 4, d(8) = 8, d(9) = 6, d(10) = 5, d(11) = 7, d(12) = 1, d(13) = 7, d(14) = 5 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 9, is 0, 0, 8, 1, 7, 2, 6, 3, 1, 0, 1, 2, 6, 0, 7, 3, 4, 4, 8, 3, 1, 0, 1, 4, 8, 6, 4, 6, 7, 4, 8, 0, 7, 3, 4, 1, 5, 3, 1, 0, 1, 1, 5, 6, 8, 1, 1, 4, 2, 6, 2, 7, 1, 8, 0, 0, 8, 1, 7, 2, 6, 3, 4, 3, 1, 2, 6, 0, 1, 6, 4, 1, 5, 3, 4, 3, 1, 1, 5, 6, 4, 6, 7, 1, 5, 0, 7, 3, 4, 7, 2, 3, 1, 0, 1, 7, 2, 6, 8, 1, 1 ----------------------------------------------------------------------------\ ----------------- This took, 0.027, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 11 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 11, (11 n + 10) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 11 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 11 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(11 n + 10) Define , 19, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1, C[3] = 3, C[4] = 3, C[5] = 6, C[6] = 6, C[7] = 10, C[8] = 10, C[9] = 4, C[10] = 4, C[11] = 10, C[12] = 10, C[13] = 6, C[14] = 6, C[15] = 3, C[16] = 3, C[17] = 1, C[18] = 1 If j is larger then, 18, then C[j]=0 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 11 n if n modulo, 2, is , 0, then Q(n) equals , ---- + 5 2 11 n if n modulo, 2, is , 1, then Q(n) equals , ---- + 11/2 2 d(n) modulo, 11, can be computed VERY fast (in logarithmic time!) using the f\ ollowing recurrence, taken modulo, 11 d(11 n + a) = Q(11 n + a) + C[a] d(n) + C[a + 11] d(n - 1) subject to the initial condition d(0) = 5, d(1) = 5, d(2) = 10, d(3) = 4, d(4) = 9, d(5) = 8, d(6) = 2, d(7) = 6, d(8) = 0, d(9) = 9, d(10) = 3, d(11) = 6, d(12) = 5, d(13) = 2, d(14) = 6, d(15) = 8, d(16) = 6, d(17) = 2, d(18) = 5 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 11, is 1, 9, 0, 9, 1, 0, 4, 4, 9, 10, 0, 2, 10, 2, 6, 10, 10, 4, 0, 6, 9, 10, 9, 5, 0, 2, 4, 1, 10, 2, 7, 5, 8, 1, 2, 1, 0, 5, 2, 2, 8, 3, 7, 9, 0, 9, 9, 3, 1, 2, 9, 6, 0, 4, 5, 5, 2, 9, 2, 5, 5, 4, 0, 6, 9, 8, 7, 10, 5, 1, 3, 3, 1, 5, 10, 7, 2, 3, 10, 4, 1, 10, 8, 10, 9, 4, 4, 7, 8, 8, 10, 7, 10, 4, 8, 10, 4, 3, 10, 6, 4 ----------------------------------------------------------------------------\ ----------------- This took, 0.030, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 12 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 12, (12 n + 11) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 12 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 12 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(12 n + 11) Define , 21, integers as follows: C[0] = 0, C[1] = 1, C[2] = 3, C[3] = 7, C[4] = 1, C[5] = 10, C[6] = 10, C[7] = 2, C[8] = 10, C[9] = 11, C[10] = 5, C[11] = 5, C[12] = 11, C[13] = 10, C[14] = 2, C[15] = 10, C[16] = 10, C[17] = 1, C[18] = 7, C[19] = 3, C[20] = 1 If j is larger then, 20, then C[j]=0 Also let Q(n) be the polynomial, 6 n + 6 d(n) modulo, 12, can be computed VERY fast (in logarithmic time!) using the f\ ollowing recurrence, taken modulo, 12 d(12 n + a) = Q(12 n + a) + C[a] d(n) + C[a + 12] d(n - 1) subject to the initial condition d(0) = 6, d(1) = 6, d(2) = 0, d(3) = 6, d(4) = 0, d(5) = 0, d(6) = 6, d(7) = 0, d(8) = 6, d(9) = 6, d(10) = 0, d(11) = 6, d(12) = 0, d(13) = 6, d(14) = 0, d(15) = 6, d(16) = 0, d(17) = 6, d(18) = 0, d(19) = 6, d(20) = 0 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 12, is 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 0, 6, 0, 6 ----------------------------------------------------------------------------\ ----------------- This took, 0.048, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 13 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 13, (13 n + 12) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 13 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 13 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(13 n + 12) Define , 23, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1, C[3] = 3, C[4] = 3, C[5] = 6, C[6] = 6, C[7] = 10, C[8] = 10, C[9] = 2, C[10] = 2, C[11] = 8, C[12] = 8, C[13] = 2, C[14] = 2, C[15] = 10, C[16] = 10, C[17] = 6, C[18] = 6, C[19] = 3, C[20] = 3, C[21] = 1, C[22] = 1 If j is larger then, 22, then C[j]=0 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 13 n if n modulo, 2, is , 0, then Q(n) equals , ---- + 6 2 13 n if n modulo, 2, is , 1, then Q(n) equals , ---- + 13/2 2 d(n) modulo, 13, can be computed VERY fast (in logarithmic time!) using the f\ ollowing recurrence, taken modulo, 13 d(13 n + a) = Q(13 n + a) + C[a] d(n) + C[a + 13] d(n - 1) subject to the initial condition d(0) = 6, d(1) = 6, d(2) = 12, d(3) = 5, d(4) = 11, d(5) = 10, d(6) = 3, d(7) = 8, d(8) = 1, d(9) = 12, d(10) = 5, d(11) = 9, d(12) = 2, d(13) = 12, d(14) = 11, d(15) = 1, d(16) = 6, d(17) = 2, d(18) = 0, d(19) = 2, d(20) = 6, d(21) = 1, d(22) = 11 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 13, is 6, 1, 8, 0, 0, 2, 8, 7, 2, 11, 8, 1, 0, 3, 4, 4, 7, 4, 9, 3, 10, 8, 4, 6, 4, 10, 10, 7, 9, 10, 1, 6, 12, 8, 0, 0, 4, 8, 11, 6, 8, 7, 8, 11, 11, 5, 4, 10, 8, 0, 10, 1, 10, 0, 8, 10, 4, 5, 11, 11, 1, 0, 0, 11, 8, 5, 12, 8, 12, 7, 8, 2, 0, 2, 10, 7, 12, 4, 6, 6, 5, 0, 9, 12, 5, 3, 5, 11, 9, 10, 4, 0, 3, 7, 6, 5, 0, 7, 11, 10, 10 ----------------------------------------------------------------------------\ ----------------- This took, 0.074, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 15 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 15, (15 n + 14) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 15 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 15 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(15 n + 14) Define , 27, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1, C[3] = 3, C[4] = 3, C[5] = 6, C[6] = 6, C[7] = 10, C[8] = 10, C[9] = 0, C[10] = 0, C[11] = 6, C[12] = 6, C[13] = 13, C[14] = 13, C[15] = 6, C[16] = 6, C[17] = 0, C[18] = 0, C[19] = 10, C[20] = 10, C[21] = 6, C[22] = 6, C[23] = 3, C[24] = 3, C[25] = 1, C[26] = 1 If j is larger then, 26, then C[j]=0 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 15 n if n modulo, 2, is , 0, then Q(n) equals , ---- + 7 2 15 n if n modulo, 2, is , 1, then Q(n) equals , ---- + 15/2 2 d(n) modulo, 15, can be computed VERY fast (in logarithmic time!) using the f\ ollowing recurrence, taken modulo, 15 d(15 n + a) = Q(15 n + a) + C[a] d(n) + C[a + 15] d(n - 1) subject to the initial condition d(0) = 7, d(1) = 7, d(2) = 14, d(3) = 6, d(4) = 13, d(5) = 12, d(6) = 4, d(7) = 10, d(8) = 2, d(9) = 0, d(10) = 7, d(11) = 12, d(12) = 4, d(13) = 1, d(14) = 8, d(15) = 12, d(16) = 11, d(17) = 7, d(18) = 13, d(19) = 1, d(20) = 14, d(21) = 9, d(22) = 14, d(23) = 1, d(24) = 13, d(25) = 7, d(26) = 11 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 15, is 5, 6, 0, 14, 1, 13, 8, 3, 6, 14, 10, 3, 7, 0, 12, 5, 10, 3, 1, 9, 0, 8, 7, 0, 7, 0, 13, 6, 10, 3, 13, 6, 7, 0, 7, 0, 13, 6, 10, 3, 13, 8, 9, 6, 13, 12, 10, 11, 0, 3, 13, 3, 4, 11, 3, 12, 8, 4, 4, 2, 6, 6, 14, 1, 13, 2, 3, 9, 14, 7, 1, 13, 11, 12, 14, 4, 10, 4, 14, 12, 11, 13, 1, 7, 14, 9, 3, 2, 13, 1, 14, 6, 6, 2, 4, 4, 8, 12, 3, 11, 4 ----------------------------------------------------------------------------\ ----------------- This took, 0.066, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 16 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 16, (16 n + 15) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 16 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 16 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(16 n + 15) Define , 29, integers as follows: C[0] = 0, C[1] = 1, C[2] = 3, C[3] = 7, C[4] = 13, C[5] = 6, C[6] = 2, C[7] = 2, C[8] = 6, C[9] = 15, C[10] = 13, C[11] = 1, C[12] = 11, C[13] = 12, C[14] = 4, C[15] = 4, C[16] = 12, C[17] = 11, C[18] = 1, C[19] = 13, C[20] = 15, C[21] = 6, C[22] = 2, C[23] = 2, C[24] = 6, C[25] = 13, C[26] = 7, C[27] = 3, C[28] = 1 If j is larger then, 28, then C[j]=0 Also let Q(n) be the polynomial, 8 n + 8 d(n) modulo, 16, can be computed VERY fast (in logarithmic time!) using the f\ ollowing recurrence, taken modulo, 16 d(16 n + a) = Q(16 n + a) + C[a] d(n) + C[a + 16] d(n - 1) subject to the initial condition d(0) = 8, d(1) = 8, d(2) = 0, d(3) = 8, d(4) = 0, d(5) = 0, d(6) = 8, d(7) = 0, d(8) = 8, d(9) = 8, d(10) = 0, d(11) = 8, d(12) = 0, d(13) = 0, d(14) = 8, d(15) = 0, d(16) = 8, d(17) = 0, d(18) = 8, d(19) = 0, d(20) = 8, d(21) = 0, d(22) = 8, d(23) = 0, d(24) = 8, d(25) = 0, d(26) = 8, d(27) = 0, d(28) = 8 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 16, is 8, 8, 0, 8, 0, 0, 8, 0, 8, 8, 0, 8, 0, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 8, 0, 8, 0, 0, 8, 0, 8, 8, 0, 8, 0, 0, 8, 0, 8, 8, 0, 8, 0, 0, 8, 0, 8, 8, 0, 8, 0, 0, 8, 0, 8, 8, 0, 8, 0, 0, 8, 0, 8, 8, 0, 8, 0, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 8, 0, 8, 0 ----------------------------------------------------------------------------\ ----------------- This took, 0.059, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 17 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 17, (17 n + 16) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 17 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 17 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(17 n + 16) Define , 31, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1, C[3] = 3, C[4] = 3, C[5] = 6, C[6] = 6, C[7] = 10, C[8] = 10, C[9] = 15, C[10] = 15, C[11] = 4, C[12] = 4, C[13] = 11, C[14] = 11, C[15] = 2, C[16] = 2, C[17] = 11, C[18] = 11, C[19] = 4, C[20] = 4, C[21] = 15, C[22] = 15, C[23] = 10, C[24] = 10, C[25] = 6, C[26] = 6, C[27] = 3, C[28] = 3, C[29] = 1, C[30] = 1 If j is larger then, 30, then C[j]=0 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 17 n if n modulo, 2, is , 0, then Q(n) equals , ---- + 8 2 17 n if n modulo, 2, is , 1, then Q(n) equals , ---- + 17/2 2 d(n) modulo, 17, can be computed VERY fast (in logarithmic time!) using the f\ ollowing recurrence, taken modulo, 17 d(17 n + a) = Q(17 n + a) + C[a] d(n) + C[a + 17] d(n - 1) subject to the initial condition d(0) = 8, d(1) = 8, d(2) = 16, d(3) = 7, d(4) = 15, d(5) = 14, d(6) = 5, d(7) = 12, d(8) = 3, d(9) = 1, d(10) = 9, d(11) = 15, d(12) = 6, d(13) = 3, d(14) = 11, d(15) = 16, d(16) = 7, d(17) = 3, d(18) = 2, d(19) = 6, d(20) = 13, d(21) = 8, d(22) = 6, d(23) = 9, d(24) = 15, d(25) = 9, d(26) = 6, d(27) = 8, d(28) = 13, d(29) = 6, d(30) = 2 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 17, is 9, 6, 10, 8, 2, 1, 2, 2, 10, 11, 14, 16, 14, 0, 10, 14, 2, 7, 7, 13, 8, 15, 5, 13, 15, 7, 4, 0, 9, 9, 13, 0, 16, 7, 1, 13, 2, 1, 2, 5, 1, 8, 16, 10, 14, 12, 12, 14, 10, 16, 8, 1, 6, 3, 4, 5, 2, 7, 0, 9, 15, 10, 12, 10, 8, 9, 3, 7, 14, 4, 7, 0, 16, 12, 7, 6, 14, 16, 5, 10, 14, 5, 7, 1, 1, 15, 13, 13, 9, 12, 6, 12, 4, 13, 3, 15, 3, 1, 4, 5, 6 ----------------------------------------------------------------------------\ ----------------- This took, 0.083, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 19 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 19, (19 n + 18) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 19 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 19 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(19 n + 18) Define , 35, integers as follows: C[0] = 0, C[1] = 1, C[2] = 1, C[3] = 3, C[4] = 3, C[5] = 6, C[6] = 6, C[7] = 10, C[8] = 10, C[9] = 15, C[10] = 15, C[11] = 2, C[12] = 2, C[13] = 9, C[14] = 9, C[15] = 17, C[16] = 17, C[17] = 7, C[18] = 7, C[19] = 17, C[20] = 17, C[21] = 9, C[22] = 9, C[23] = 2, C[24] = 2, C[25] = 15, C[26] = 15, C[27] = 10, C[28] = 10, C[29] = 6, C[30] = 6, C[31] = 3, C[32] = 3, C[33] = 1, C[34] = 1 If j is larger then, 34, then C[j]=0 Also let Q(n) be the quasi-polynomial of quasi-period, 2, such that 19 n if n modulo, 2, is , 0, then Q(n) equals , ---- + 9 2 19 n if n modulo, 2, is , 1, then Q(n) equals , ---- + 19/2 2 d(n) modulo, 19, can be computed VERY fast (in logarithmic time!) using the f\ ollowing recurrence, taken modulo, 19 d(19 n + a) = Q(19 n + a) + C[a] d(n) + C[a + 19] d(n - 1) subject to the initial condition d(0) = 9, d(1) = 9, d(2) = 18, d(3) = 8, d(4) = 17, d(5) = 16, d(6) = 6, d(7) = 14, d(8) = 4, d(9) = 2, d(10) = 11, d(11) = 18, d(12) = 8, d(13) = 5, d(14) = 14, d(15) = 1, d(16) = 10, d(17) = 6, d(18) = 15, d(19) = 1, d(20) = 0, d(21) = 14, d(22) = 3, d(23) = 7, d(24) = 5, d(25) = 18, d(26) = 6, d(27) = 9, d(28) = 6, d(29) = 18, d(30) = 5, d(31) = 7, d(32) = 3, d(33) = 14, d(34) = 0 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 19, is 13, 7, 6, 6, 14, 1, 18, 11, 18, 17, 14, 0, 6, 17, 13, 11, 16, 1, 15, 6, 10, 7, 1, 4, 7, 16, 9, 5, 7, 9, 1, 9, 10, 5, 15, 16, 16, 4, 13, 7, 14, 14, 0, 6, 9, 2, 3, 2, 1, 6, 3, 14, 9, 7, 0, 4, 14, 5, 13, 12, 18, 6, 10, 6, 8, 12, 12, 5, 3, 4, 0, 9, 3, 1, 12, 18, 8, 3, 1, 4, 10, 2, 16, 16, 0, 8, 0, 16, 16, 2, 10, 4, 1, 3, 8, 18, 12, 15, 17, 13, 4 ----------------------------------------------------------------------------\ ----------------- This took, 0.131, seconds, ----------------------------------------------------------------------------\ ----------------- m is, 20 ----------------------------------------------------------------------------\ ----------------- A Fast (Log Time!) Way to Determine the Remainder Upon Dividing by, 20, (20 n + 19) of the coefficient of, q in the Unique Formal Power Series Satisfying the Functional Equation 20 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) By Shalosh B. Ekhad Theorem: Let c(n) be the coefficient of q^n in the formal power series of th\ e title, in other words let infinity ----- \ n f(q) = ) c(n) q / ----- n = 0 is the unique formal power series satisfying the inhomogeneneous functional \ equation: 20 q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) Let , d(n), be , c(20 n + 19) Define , 37, integers as follows: C[0] = 0, C[1] = 1, C[2] = 3, C[3] = 7, C[4] = 13, C[5] = 2, C[6] = 14, C[7] = 10, C[8] = 10, C[9] = 15, C[10] = 5, C[11] = 1, C[12] = 3, C[13] = 12, C[14] = 8, C[15] = 12, C[16] = 4, C[17] = 5, C[18] = 15, C[19] = 15, C[20] = 5, C[21] = 4, C[22] = 12, C[23] = 8, C[24] = 12, C[25] = 3, C[26] = 1, C[27] = 5, C[28] = 15, C[29] = 10, C[30] = 10, C[31] = 14, C[32] = 2, C[33] = 13, C[34] = 7, C[35] = 3, C[36] = 1 If j is larger then, 36, then C[j]=0 Also let Q(n) be the polynomial, 10 n + 10 d(n) modulo, 20, can be computed VERY fast (in logarithmic time!) using the f\ ollowing recurrence, taken modulo, 20 d(20 n + a) = Q(20 n + a) + C[a] d(n) + C[a + 20] d(n - 1) subject to the initial condition d(0) = 10, d(1) = 10, d(2) = 0, d(3) = 10, d(4) = 0, d(5) = 0, d(6) = 10, d(7) = 0, d(8) = 10, d(9) = 10, d(10) = 0, d(11) = 10, d(12) = 0, d(13) = 0, d(14) = 10, d(15) = 0, d(16) = 10, d(17) = 10, d(18) = 0, d(19) = 10, d(20) = 0, d(21) = 10, d(22) = 0, d(23) = 10, d(24) = 0, d(25) = 10, d(26) = 0, d(27) = 10, d(28) = 0, d(29) = 10, d(30) = 0, d(31) = 10, d(32) = 0, d(33) = 10, d(34) = 0, d(35) = 10, d(36) = 0 Just for fun the googol-th through googol+100-th terms of d(n) modulo, 20, is 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0 ----------------------------------------------------------------------------\ ----------------- This took, 0.113, seconds, ----------------------------------------------------------------------------\ ----------------- This took, 0.777, seconds.