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

0,2

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

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

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).

Example

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.

Mathematica

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

Prog

(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))};
(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

Crossrefs

Cf. A227595.

Cf. A000217.

Sequence in context: A073135 A063804 A213800 * A372387 A078753 A119443

Adjacent sequences: A224820 A224821 A224822 * A224824 A224825 A224826

Keyword

nonn

Author

Michael Somos, Jul 20 2013

Status

approved