On the number of partitions into parts not congruent to \(0, \pm 3\pmod{12}\)
Let \(R(n)\) denote the number of partitions of the positive integer \(n\) into parts not congruent to \(0, \pm 3\pmod{12}\). \textit{A. V. Sills} [Ramanujan J. 23, 253--264 (2010; Zbl 1211.11112)] gave a Rademacher-type formula for \(R(n)\). Computing \(R(n)\) by this formula requires arithmetic with very high-precision approximate real numbers. As to details on how to efficiently implement a Rademacher-type formula see \textit{F. Johansson} [LMS J. Comput. Math. 15, 341--359 (2012; Zbl 1344.11089)].\par In the paper under review the author investigates other relations for computing the value of \(R(n)\). His Theorem 2.1 yields Corollary 2.2 which is a linear recurrence relation for \(R(n)\) involving the generalized pentagonal numbers. Theorem 3.1 yields Corollary 3.2, the following more efficient recurrence relation for computing \(R(n)\): For \(n\ge 0\), \[R(n)+2\sum_{j=1}^{\infty}(-1)^j R(n-2j^2)=\begin{cases} 1, \;\text{if}\; n=k(3k-2)\\0, \;\text{otherwise.}\end{cases}\] The author also derives that \(R(n)\) is odd if and only if \(3n+1\) is a square. According to the proof of Corollary 5.1, \[R(n)=\sum_{k=0}^{\infty}(-1)^{k(k+1)/2}p(n-3k(k+1)/2)\] with the ordinary partition function \(p(n)\), therefore, for \(n\ge 0\), \[\sum_{8k+1\; \text{square}}p(n-3k)\equiv 1 \bmod 2 \] if and only if \(3n+1\) is a square.
scientific article; zbMATH DE number 7434710