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

%I A133827 #26 Feb 16 2025 08:33:06
%S A133827 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,
%T A133827 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,
%U A133827 2,0,0,0,0,0,2,2,0,0,0,0,2,2,0,0,1,0,0
%N A133827 Number of solutions to x + 7 * y = 2 * n in triangular numbers.
%C A133827 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A133827 G.f. is called omega(q) by Berkovich and Yesilyurt.
%H A133827 G. C. Greubel, <a href="/A133827/b133827.txt">Table of n, a(n) for n = 0..10000</a>
%H A133827 Alexander Berkovich and Hamza Yesilyurt, <a href="https://arxiv.org/abs/math/0603150">New Identities For 7-cores with Prescribed BG-Rank</a>, arXiv:math/0603150 [math.NT], 2006-2007, page 3, equation (1.19).
%H A133827 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A133827 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A133827 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.
%F A133827 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).
%F A133827 a(7*n + 1) = a(7*n + 2) = a(7*n + 6) = 0. a(7*n + 3) = a(n).
%F A133827 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.
%F A133827 a(n) = A035162(2*n + 1) = A035182(2*n + 1) = A110399(2*n + 1) = A121454(2*n + 1).
%F A133827 2 * a(n) = A002652(2*n + 1) = A033719(2*n + 1). - _Michael Somos_, Dec 30 2016
%F A133827 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(7)) = 0.593705... . - _Amiram Eldar_, Dec 29 2023
%e A133827 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 + ...
%e A133827 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 + ...
%t A133827 a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[#, 2] KroneckerSymbol[ -28, #] &]]; (* _Michael Somos_, Oct 30 2015 *)
%t A133827 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 *)
%o A133827 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, (d%2) * kronecker( -28, d)))};
%o A133827 (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)))};
%Y A133827 Cf. A002652, A033719, A035162, A035182, A110399, A121454.
%Y A133827 Cf. A000122, A000700, A010054, A121373.
%K A133827 nonn,easy
%O A133827 0,6
%A A133827 _Michael Somos_, Sep 25 2007, Oct 04 2008