Congruence Properties of Binary Partition Functions

Abstract

Abstract. Let A be a finite subset of N containing 0, and let f (n) denote the number of ways to write n in the form ǫ j 2 j , where ǫ j ∈ A. We show that there exists a computable T = T (A) so that the sequence (f (n) mod 2) is periodic with period T . Variations and generalizations of this problem are also discussed.

Odd behavior in the coefficients of reciprocals of binary power series

“…In [1], Anders, Dennison, Lansing, and Reznick studied the behavior of the sequence (f A (n)) mod 2 and proved the following theorem.…”

“…Cooper, Eichhorn, and O'Bryant considered the fraction γ(f (x)) defined in, as did we in [1], but here we have instead considered the ordered pair…”

“…for various polynomials f (x), as did we in [1]. Here we instead consider the ordered pair (1.2) β(f (x)) := ( 1 (f * ), 0,D−1 (f * )), which gives more precise information than reduced fractions would.…”

Counting non-standard binary representations

“…so for any odd n, f A (n) satisfies a recurrence relation of order c t . Dennison, Lansing, Reznick, and the author [3] gave this argument for f A,b (n), the b-ary representation of n with coefficients from A, using residue classes mod b.…”