A054843 - OEIS

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

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OFFSET

0,2

COMMENTS

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

LINKS

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

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

EXAMPLE

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.

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

PROG

(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

CROSSREFS

Cf. A001227, A010054, A054844.

Sequence in context: A279782 A358921 A132589 * A277427 A231631 A280700

Adjacent sequences: A054840 A054841 A054842 * A054844 A054845 A054846

KEYWORD

easy,nonn

AUTHOR

Henry Bottomley, Apr 13 2000

EXTENSIONS

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

STATUS

approved