OFFSET
1,3
COMMENTS
Also a(n) is the total number of ways to represent n+1 as a centered polygonal number of the form km(m+1)/2+1 for k>1. - Alexander Adamchuk, Apr 26 2007
Number of oblong numbers that divide 2n. - Ray Chandler, Jun 24 2008
The number of divisors d of 2n such that d+1 is also a divisor of 2n, see first formula. - Michel Marcus, Jun 18 2015
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n forming a finite arithmetic progression with offset 0, i.e. the differences are all equal (with the last part taken to be 0). The Heinz numbers of these partitions are given by A325327. For example, the a(1) = 1 through a(12) = 3 partitions are (A = 10, B = 11, C = 12):
1 2 3 4 5 6 7 8 9 A B C
21 42 63 4321 84
321 642
(End)
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number.
Wikipedia, Arithmetic progression.
FORMULA
a(n) = Sum_{d|2*n,d+1|2*n} 1.
a(n) = A129308(2n). - Ray Chandler, Jun 24 2008
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Dec 31 2023
MATHEMATICA
sup=90; TriN=Array[ (#+1)(#+2)/2&, Floor[ N[ Sqrt[ sup*2 ] ] ]-1 ]; Array[ Function[n, 1+Count[ Map[ Mod[ n, # ]&, TriN ], 0 ] ], sup ]
Table[Count[Divisors[k], _?(IntegerQ[Sqrt[8 # + 1]] &)], {k, 105}] (* Jayanta Basu, Aug 12 2013 *)
Table[Length[Select[IntegerPartitions[n], SameQ@@Differences[Append[#, 0]]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *)
PROG
(Haskell)
a007862 = sum . map a010054 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014
(PARI) a(n) = sumdiv(n, d, ispolygonal(d, 3)); \\ Michel Marcus, Jun 18 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Extended by Ray Chandler, Jun 24 2008
STATUS
approved