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 A224823 Number of solutions to n = x + y + 3*z where x, y, z are triangular numbers. 5

%I

%S 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 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 14,14,0,7,16,7,18,14,4,19,10,0,12,16,9,13,20,0

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

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C a(A224829(n)) = 0. - _Reinhard Zumkeller_, Jul 21 2013

%H Reinhard Zumkeller, <a href="/A224823/b224823.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of psi(x)^2 * psi(x^3) in powers of x where psi() is a Ramanujan theta function.

%F Expansion of q^(-5/8) * eta(q^2)^4 * eta(q^6)^2 / (eta(q)^2 * eta(q^3)) in powers of q.

%F Euler transform of period 6 sequence [ 2, -2, 3, -2, 2, -3, ...].

%F G.f.: (Sum_{k>0} x^((k^2-k)/2))^2 * (Sum_{k>0} x^(3 * (k^2-k)/2)).

%F -2 * a(n) = A227595(3*n + 1).

%e 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 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 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 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 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 2, 0, x^(3/2)] / (8 x^(5/8)), {x, 0, n}];

%o (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 n = length [() | let ts = takeWhile (<= n) a000217_list,

%o x <- ts, y <- ts, z <- takeWhile (<= div (n - x - y) 3) ts,

%o x + y + 3 * z == n]

%o -- _Reinhard Zumkeller_, Jul 21 2013

%Y Cf. A227595.

%Y Cf. A000217.

%K nonn

%O 0,2

%A _Michael Somos_, Jul 20 2013

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Last modified May 19 11:12 EDT 2019. Contains 323390 sequences. (Running on oeis4.)