A117571 - OEIS

A117571

Expansion of (1+2*x^2)/((1-x)*(1-x^3)).

1, 1, 3, 4, 4, 6, 7, 7, 9, 10, 10, 12, 13, 13, 15, 16, 16, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 28, 30, 31, 31, 33, 34, 34, 36, 37, 37, 39, 40, 40, 42, 43, 43, 45, 46, 46, 48, 49, 49, 51, 52, 52, 54, 55, 55, 57, 58, 58, 60, 61, 61, 63, 64, 64, 66, 67, 67, 69, 70, 70, 72

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Row sums of A116948.

Place n+2 equally-spaced points around a circle, labeled 0,1,2,...,n+1. For each i = 0..n+1 such that 2i != i mod n+2, draw an (undirected) chord from i to 2i mod n+2. Then a(n) is the number of distinct chords. - Kival Ngaokrajang, May 13 2016 (Edited by N. J. A. Sloane, Jun 23 2016)

From Gus Wiseman, Apr 19 2019: (Start)

Also the number of integer partitions of n + 2 with 1 fewer distinct multiplicities than (not necessarily distinct) parts. These are partitions of the form (x,x), (x,y), (x,x,y), or (x,y,y). For example, the a(0) = 1 through a(8) = 9 partitions are the following. The Heinz numbers of these partitions are given by A325270.

(11) (21) (22) (32) (33) (43) (44) (54) (55)

(31) (41) (42) (52) (53) (63) (64)

(211) (221) (51) (61) (62) (72) (73)

(311) (411) (322) (71) (81) (82)

(331) (332) (441) (91)

(511) (422) (522) (433)

(611) (711) (442)

(622)

(811)

(End)

LINKS

Table of n, a(n) for n=0..71.