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

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

Offset: 0

Author

Michael Somos, Jul 20 2013

Keywords

• nonn

Comments

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

a(A224829(n)) = 0. - Reinhard Zumkeller, Jul 21 2013

Examples

			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 + ...
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 + ...
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.
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.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A227595.

Cf. A000217.

Programs

Haskell

a224823 n = length [() | let ts = takeWhile (<= n) a000217_list,
            x <- ts, y <- ts, z <- takeWhile (<= div (n - x - y) 3) ts,
            x + y + 3 * z == n]
-- Reinhard Zumkeller, Jul 21 2013

Mathematica

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

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))};

Formula

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

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

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

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

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