A054843 Number of sequences of consecutive nonnegative integers (including sequences of length 1) that sum to n.

1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 3, 2, 2, 2, 2, 5, 1, 2, 3, 2, 2, 5, 2, 2, 2, 3, 2, 4, 3, 2, 4, 2, 1, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 2, 2, 7, 2, 2, 2, 3, 3, 4, 2, 2, 4, 5, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 5, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 5, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 5, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 9

Offset: 0

Author

Henry Bottomley, Apr 13 2000

Keywords

• easy
• nonn

Comments

Number of nonnegative integer solutions (x, y) to the equation (2*x + y)*(1 + y)/2 = n. - Gionata Neri, Nov 15 2015

Examples

			a(0) = 1 because 0 = 0;
a(1) = 2 because 1 = 0+1 or 1;
a(15) = 5 because 15 = 0+1+2+3+4+5 or 1+2+3+4+5 or 4+5+6 or 7+8 or 15.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001227, A010054, A054844.

Programs

Maple

N:= 1000:
G:= add(x^(k*(k-1)/2)/(1-x^k),k=1..floor((1+sqrt(1+8*N))/2)):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Nov 15 2015

PARI

vector(100, n, local(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x^2+A)^2/eta(x+A), n)) + numdiv(n>>valuation(n, 2))) \\ Altug Alkan, Nov 15 2015

Formula

From Vladeta Jovovic, Aug 10 2004: (Start)

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

a(n) = A001227(n) + A010054(n), for n>0. (End)

a(2^k) = 1 for k > 0. - Daniel Castle, Feb 09 2021

Extensions

a(0) = 1 added by N. J. A. Sloane, Dec 02 2020