cp's OEIS Frontend

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

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