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A224823 Number of solutions to n = x + y + 3*z where x, y, z are triangular numbers.

%I A224823 #18 Feb 16 2025 08:33:19
%S A224823 1,2,1,3,4,1,5,4,0,6,6,3,5,6,2,6,8,0,7,8,4,9,6,1,11,10,0,8,6,5,9,12,3,
%T A224823 7,14,0,11,8,5,13,10,4,8,8,0,14,16,5,11,12,1,16,10,0,14,14,7,9,12,5,
%U A224823 14,14,0,7,16,7,18,14,4,19,10,0,12,16,9,13,20,0
%N A224823 Number of solutions to n = x + y + 3*z where x, y, z are triangular numbers.
%C A224823 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A224823 a(A224829(n)) = 0. - _Reinhard Zumkeller_, Jul 21 2013
%H A224823 Reinhard Zumkeller, <a href="/A224823/b224823.txt">Table of n, a(n) for n = 0..10000</a>
%H A224823 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A224823 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A224823 Expansion of psi(x)^2 * psi(x^3) in powers of x where psi() is a Ramanujan theta function.
%F A224823 Expansion of q^(-5/8) * eta(q^2)^4 * eta(q^6)^2 / (eta(q)^2 * eta(q^3)) in powers of q.
%F A224823 Euler transform of period 6 sequence [ 2, -2, 3, -2, 2, -3, ...].
%F A224823 G.f.: (Sum_{k>0} x^((k^2-k)/2))^2 * (Sum_{k>0} x^(3 * (k^2-k)/2)).
%F A224823 -2 * a(n) = A227595(3*n + 1).
%e A224823 G.f. = 1 + 2*x + x^2 + 3*x^3 + 4*x^4 + x^5 + 5*x^6 + 4*x^7 + 6*x^9 + 6*x^10 + ...
%e A224823 G.f. = q^5 + 2*q^13 + q^21 + 3*q^29 + 4*q^37 + q^45 + 5*q^53 + 4*q^61 + 6*q^77 + ...
%e A224823 a(3) = 3 since 3 = 0 + 0 + 3*1 = 0 + 3 + 3*0 = 3 + 0 + 3*0 are the 3 solutions of 3 = x + y + 3*z in triangular numbers.
%e A224823 a(4) = 4 since 4 = 1 + 0 + 3*1 = 0 + 1 + 3*1 = 3 + 1 + 3*0 = 1 + 3 + 3*0 are the 4 solutions of 4 = x + y + 3*z in triangular numbers.
%t A224823 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 2, 0, x^(3/2)] / (8 x^(5/8)), {x, 0, n}];
%o A224823 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^3 + A)), n))};
%o A224823 (Haskell)
%o A224823 a224823 n = length [() | let ts = takeWhile (<= n) a000217_list,
%o A224823             x <- ts, y <- ts, z <- takeWhile (<= div (n - x - y) 3) ts,
%o A224823             x + y + 3 * z == n]
%o A224823 -- _Reinhard Zumkeller_, Jul 21 2013
%Y A224823 Cf. A227595.
%Y A224823 Cf. A000217.
%K A224823 nonn
%O A224823 0,2
%A A224823 _Michael Somos_, Jul 20 2013