A023361 Number of compositions of n into positive triangular numbers.

1, 1, 1, 2, 3, 4, 7, 11, 16, 25, 40, 61, 94, 147, 227, 351, 546, 846, 1309, 2030, 3147, 4876, 7558, 11715, 18154, 28136, 43609, 67586, 104748, 162346, 251610, 389958, 604381, 936699, 1451743, 2249991, 3487153, 5404570, 8376292, 12982016, 20120202, 31183350

Offset: 0

Author

David W. Wilson, Jun 14 1998

Keywords

• nonn

Comments

Number of compositions [c(1), c(2), c(3), ...] of n such that either c(k) = c(k-1) + 1 or c(k) = 1; see example. Same as fountains of coins (A005169) where each valley is at the lowest level. - Joerg Arndt, Mar 25 2014

Examples

			From _Joerg Arndt_, Mar 25 2014: (Start)
There are a(9) = 25 compositions of 9 such that either c(k) = c(k-1) + 1 or c(k) = 1:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 2 1 1 1 ]
06:  [ 1 1 1 1 2 1 2 ]
07:  [ 1 1 1 1 2 3 ]
08:  [ 1 1 1 2 1 1 1 1 ]
09:  [ 1 1 1 2 1 1 2 ]
10:  [ 1 1 1 2 1 2 1 ]
11:  [ 1 1 1 2 3 1 ]
12:  [ 1 1 2 1 1 1 1 1 ]
13:  [ 1 1 2 1 1 1 2 ]
14:  [ 1 1 2 1 1 2 1 ]
15:  [ 1 1 2 1 2 1 1 ]
16:  [ 1 1 2 3 1 1 ]
17:  [ 1 2 1 1 1 1 1 1 ]
18:  [ 1 2 1 1 1 1 2 ]
19:  [ 1 2 1 1 1 2 1 ]
20:  [ 1 2 1 1 2 1 1 ]
21:  [ 1 2 1 2 1 1 1 ]
22:  [ 1 2 1 2 1 2 ]
23:  [ 1 2 1 2 3 ]
24:  [ 1 2 3 1 1 1 ]
25:  [ 1 2 3 1 2 ]
The last few, together with the corresponding fountains of coins are:
.  20:  [ 1 2 1 1 2 1 1 ]
.
.     O     O
.    O O O O O O O
.
.
.  21:  [ 1 2 1 2 1 1 1 ]
.
.     O   O
.    O O O O O O O
.
.
.  22:  [ 1 2 1 2 1 2 ]
.
.     O   O   O
.    O O O O O O
.
.
.  23:  [ 1 2 1 2 3 ]
.
.           O
.      O   O O
.     O O O O O
.
.
.  24:  [ 1 2 3 1 1 1 ]
.
.       O
.      O O
.     O O O O O O
.
.
.  25:  [ 1 2 3 1 2 ]
.
.       O
.      O O   O
.     O O O O O
(End)
Applying recursion formula: 40 = a(10) = a(9) + a(7) + a(4) + a(0) = 25 + 11 + 3 + 1. - _Gregory L. Simay_, Jun 14 2016
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5256 (terms n = 0..500 from T. D. Noe)
• N. Robbins, On compositions whose parts are polygonal numbers, Annales Univ. Sci. Budapest., Sect. Comp. 43 (2014) 239-243. See p. 242.
• Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Crossrefs

Cf. A106332.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(`if`(issqr(8*j+1), a(n-j), 0), j=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2017

Mathematica

(1/(2 - QPochhammer[x^2]/QPochhammer[x, x^2]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 23 2016 *)
a[n_] := a[n] = If[n == 0, 1, Sum[ If[ IntegerQ[ Sqrt[8j+1]], a[n-j], 0], {j, 1, n}]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

PARI

N=66;  x='x+O('x^N);
Vec( 1/( 1 - sum(k=1,1+sqrtint(2*N), x^binomial(k+1,2) ) ) )
/* Joerg Arndt, Sep 30 2012 */

Formula

G.f. : 1 / (1 - Sum_{k>=1} x^(k*(k+1)/2) ).

a(n) ~ c * d^n, where d = 1/A106332 = 1.5498524695188884304192160776463163555... is the root of the equation d^(1/8) * EllipticTheta(2, 0, 1/sqrt(d)) = 4 and c = 0.492059962414480455851222791075288170662444559041717451009563731799... - Vaclav Kotesovec, May 01 2014, updated Feb 17 2017

a(n) = a(n-1) + a(n-3) + a(n-6) + a(n-10) + ... Gregory L. Simay, Jun 09 2016

G.f.: 1/(2 - (x^2;x^2)inf/(x;x^2)_inf), where (a;q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Sep 23 2016

G.f.: 1/(2 - theta_2(sqrt(q))/(2*q^(1/8))), where theta_2() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018