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

0,6

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.

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.

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

Example

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 + ...

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 *)

Prog

(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)))};

Crossrefs

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

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A326559 A083339 A216278 * A176192 A028633 A316898

Adjacent sequences: A133824 A133825 A133826 * A133828 A133829 A133830

Keyword

nonn,easy

Author

Michael Somos, Sep 25 2007, Oct 04 2008

Status

approved