A133827 Number of solutions to x + 7 * y = 2 * n in triangular numbers.

1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0

Offset: 0

Author

Michael Somos, Sep 25 2007, Oct 04 2008

Keywords

• nonn
• easy

Comments

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

G.f. is called omega(q) by Berkovich and Yesilyurt.

Examples

			G.f. = 1 + x^3 + x^4 + 2*x^5 + 2*x^11 + x^12 + 2*x^14 + 2*x^18 + 2*x^21 + x^24 + ...
G.f. = q + q^7 + q^9 + 2*q^11 + 2*q^23 + q^25 + 2*q^29 + 2*q^37 + 2*q^43 + q^49 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Alexander Berkovich and Hamza Yesilyurt, New Identities For 7-cores with Prescribed BG-Rank, arXiv:math/0603150 [math.NT], 2006-2007, page 3, equation (1.19).
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A002652, A033719, A035162, A035182, A110399, A121454.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[#, 2] KroneckerSymbol[ -28, #] &]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(7/2)], {x, 0, 2 n + 1}]; (* Michael Somos, Oct 30 2015 *)

PARI

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, (d%2) * kronecker( -28, d)))};

PARI

{a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k = 1, matsize(A)[1], [p, e] = A[k, ]; if(p == 2, 0, p == 7, 1, 1 == kronecker( -7, p), e + 1, 1-e%2)))};

Formula

Expansion of psi(x^4) * phi(x^14) + x^3 * psi(x^28) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(7^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), b(p^e) = e + 1 if p == 1, 2, 4 (mod 7).

a(7*n + 1) = a(7*n + 2) = a(7*n + 6) = 0. a(7*n + 3) = a(n).

Expansion of psi(q) * psi(q^7) - q * psi(q^2) * psi(q^14) = (psi(q) * psi(q^7) + psi(-q) * psi(-q^7)) / 2 in powers of q^2 where psi() is a Ramanujan theta function.

a(n) = A035162(2*n + 1) = A035182(2*n + 1) = A110399(2*n + 1) = A121454(2*n + 1).

2 * a(n) = A002652(2*n + 1) = A033719(2*n + 1). - Michael Somos, Dec 30 2016

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(7)) = 0.593705... . - Amiram Eldar, Dec 29 2023