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%I #32 Aug 05 2023 06:25:03
%S 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,
%T 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,
%U 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
%N Number of sequences of consecutive nonnegative integers (including sequences of length 1) that sum to n.
%C Number of nonnegative integer solutions (x, y) to the equation (2*x + y)*(1 + y)/2 = n. - _Gionata Neri_, Nov 15 2015
%H Robert Israel, <a href="/A054843/b054843.txt">Table of n, a(n) for n = 0..10000</a>
%F From _Vladeta Jovovic_, Aug 10 2004: (Start)
%F G.f.: Sum_{k >= 1} x^(k*(k-1)/2)/(1-x^k).
%F a(n) = A001227(n) + A010054(n), for n>0. (End)
%F a(2^k) = 1 for k > 0. - _Daniel Castle_, Feb 09 2021
%e a(0) = 1 because 0 = 0;
%e a(1) = 2 because 1 = 0+1 or 1;
%e 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.
%p N:= 1000:
%p G:= add(x^(k*(k-1)/2)/(1-x^k),k=1..floor((1+sqrt(1+8*N))/2)):
%p S:= series(G,x,N+1):
%p seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Nov 15 2015
%o (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
%Y Cf. A001227, A010054, A054844.
%K easy,nonn
%O 0,2
%A _Henry Bottomley_, Apr 13 2000
%E a(0) = 1 added by _N. J. A. Sloane_, Dec 02 2020
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