Chapter 10: Processes of Perception and Analysis

Section 11: Traditional Mathematics and Mathematical Formulas

[Generating functions for] 1D sequences

Generating functions that are rational always lead to sequences which after reduction modulo 2 are purely repetitive. Algebraic generating functions can also lead to nested sequences. (Note that to get only integer sequences such generating functions have to be specially chosen.) Sqrt[1-4x]/2 yields a sequence with 1's at positions 2m, as essentially obtained from the substitution system {2 -> {2, 1}, 1 -> {1, 0}, 0 -> {0, 0}}. Sqrt[(1 - 3 x)/(1 + x)]/2 yields sequence (a) on page 84. (1 + Sqrt[(1 - 3x)/(1 + x)])/(2(1 + x)) (see page 890) yields the Thue–Morse sequence. (This particular generating function satisfies the equation (1+x)3 f2 - (1+ x)2f + x == 0.) (1 - 9x)^(1/3) yields almost the Cantor set sequence from page 83. EllipticTheta[3, Pi, x]/2 gives a sequence with 1's at positions m2.

For any sequence with an algebraic generating function and thus for any nested sequence the nth element can always be expressed in terms of hypergeometric functions. For the Thue–Morse sequence the result is

1/2*(-1)n + ((-3)n*Sqrt[Pi]*Hypergeometric2F1[3/2, -n, 3/2 - n, -(1/3)])/(4*n!*Gamma[3/2 - n])

From Stephen Wolfram: A New Kind of Science [citation]