A229180 -id:A229180 - cp's OEIS frontend

A328798 Expansion of 1 / (chi(-x) * chi(-x^3)) in powers of x where chi() is a Ramanujan theta function.

1, 1, 1, 3, 3, 4, 7, 8, 10, 16, 19, 23, 33, 39, 48, 65, 77, 93, 122, 144, 173, 220, 259, 309, 384, 451, 534, 653, 764, 899, 1085, 1264, 1479, 1765, 2048, 2385, 2820, 3260, 3778, 4432, 5105, 5891, 6864, 7879, 9056, 10491, 12002, 13744, 15839, 18064, 20616, 23648

Offset: 0

Michael Somos, Oct 28 2019

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution inverse is A112175, 2nd power is A102315, 3rd power is A229180, 6th power is A123653.
f(-1 / (216 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A112175.

			G.f. = 1 + x + x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 8*x^7 + ...
G.f. = q + q^7 + q^13 + 3*q^19 + 3*q^25 + 4*q^31 + 7*q^37 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A102315, A112175, A123653, A229180.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^3, x^3], {x, 0, n}];

{a(n) = my(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)), n))};

Expansion of q^(-1/6) * eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [1, 0, 2, 0, 1, 0, ...].
G.f.: Product_{k>=1} (1 + x^k)^(-1) * (1 + x^(3*k))^(-1).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019

A229179 Number of solutions of x^2 + y^2 + z^2 == -1 (mod n) with x, y, and z in 0..n-1.

Original entry on oeis.org

1, 4, 12, 8, 30, 48, 56, 0, 108, 120, 132, 96, 182, 224, 360, 0, 306, 432, 380, 240, 672, 528, 552, 0, 750, 728, 972, 448, 870, 1440, 992, 0, 1584, 1224, 1680, 864, 1406, 1520, 2184, 0, 1722, 2688, 1892, 1056, 3240, 2208, 2256, 0, 2744, 3000, 3672, 1456

Offset: 1

José María Grau Ribas, Sep 29 2013

			As 60 = 4 * 3 * 5, a(60) = a(4) * a(3) * a(5) = 8 * (3 * (3 + 1)) * (5 * (5 + 1)) = 8 * 12 * 30 = 2880. - _David A. Corneth_, Jun 24 2018

Andrew Howroyd and David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2500 terms from Andrew Howroyd).
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Cf. A086932, A229180.

Table[Sum[ If[Mod[a^2 + b^2 + c^2 + 1, n] == 0, 1, 0], {c, 0, n - 1}, {b, 0,  n - 1}, {a, 0, n - 1}], {n, 14}]
f[p_, e_] := If[p == 2, Which[e == 1, 4, e == 2, 8, e > 2, 0], (p + 1)*p^(2*e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 18 2022 *)

a(n)={my(p=Mod(sum(i=0, n-1, x^(i^2 % n)), x^n-1)); polcoeff(lift(p^3), n-1)} \\ Andrew Howroyd, Jun 24 2018

first(n) = {my(res = vector(n)); forstep(i = 1, n, 2, f = factor(i); res[i] = 1; for(j = 1, #f~, res[i] *= f[j, 1] * (f[j, 1] + 1) * f[j, 1] ^ ((f[j, 2] - 1) << 1)); res); for(k = 1, 2, forstep(i = 1, n >> k, 2, res[i << k] = res[i] << (k+1))); res} \\ David A. Corneth, Jun 24 2018

a(8 * n) = 0; for odd prime p, a(p^k) = p^(2 * k - 1) * (p + 1); a(2) = 4, a(4) = 8. - David A. Corneth, Jun 24 2018

Sum_{k=1..n} a(k) ~ c * n^3, where c = 13/(4*Pi^2) = 0.329293... . - Amiram Eldar, Oct 18 2022

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A328798 Expansion of 1 / (chi(-x) * chi(-x^3)) in powers of x where chi() is a Ramanujan theta function.

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A229179 Number of solutions of x^2 + y^2 + z^2 == -1 (mod n) with x, y, and z in 0..n-1.

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