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A224823 Number of solutions to n = x + y + 3*z where x, y, z are triangular numbers. 5
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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 * A078753 A119443 A209413

Adjacent sequences:  A224820 A224821 A224822 * A224824 A224825 A224826

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 20 2013

STATUS

approved

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Last modified April 21 06:42 EDT 2019. Contains 322310 sequences. (Running on oeis4.)