A023917 -id:A023917 - cp's OEIS frontend

A072837 Duplicate of A023917.

1, 0, 0, 0, 0, 12, 0, 0, 30, 20, 0, 0, 30, 0, 0, 0, 0, 120, 0, 0, 132, 60, 0, 0, 90, 0, 0, 0, 0, 180

Offset: 0

dead

A072833 Numbers that are congruent to 0, 5, 8, 9 mod 12.

0, 5, 8, 9, 12, 17, 20, 21, 24, 29, 32, 33, 36, 41, 44, 45, 48, 53, 56, 57, 60, 65, 68, 69, 72, 77, 80, 81, 84, 89, 92, 93, 96, 101, 104, 105, 108, 113, 116, 117, 120, 125, 128, 129, 132, 137, 140, 141, 144, 149, 152, 153, 156, 161, 164, 165, 168, 173, 176, 177, 180, 185, 188, 189

Offset: 0

N. J. A. Sloane, Jul 25 2002

The exponents occurring in the expansion of F_6(q^2) (see Ahlgren) or, equivalently, the norms of the vectors in the A*5 lattice. - _Andrey Zabolotskiy, Oct 26 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Scott Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc. 128 (2000), 1333-1338.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A023917, A108752, A072835.

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; F[6, q_]:= ( -3*f[q, q]^5 + 5*f[q, q]^3*f[q^3, q^3]^2 + 15*f[q, q]*f[q^3, q^3]^4 + 15*f[q^3, q^3]^6/f[q, q]  )/32; cfs = CoefficientList[Series[F[6, q], {q, 0, 500}], q]; Take[Pick[Range[Length[cfs]] - 1, Sign[Abs[cfs]], 1], 50] (* G. C. Greubel, Apr 16 2018 *)
Flatten[#+{0,5,8,9}&/@(12*Range[0,20])] (* Harvey P. Dale, Apr 10 2022 *)

G.f.: x*(3*x^2-2*x+5) / ((x-1)^2*(x^2+1)). - Colin Barker, Jul 31 2013
Sum_{n>=1} 1/a(n) = Pi*(3-2*sqrt(3))/72 + log(2)/2 - arccoth(sqrt(3))/(2*sqrt(3)). - Amiram Eldar, Jul 26 2024
E.g.f.: exp(x)*(1 + 3*x) - cos(x) + sin(x). - Stefano Spezia, Oct 27 2024

Terms a(33) onward added by G. C. Greubel, Apr 16 2018

Edited by Andrey Zabolotskiy, Aug 14 2020

A125564 Theta series of 5-dimensional lattice A_5^{+3}.

Original entry on oeis.org

1, 0, 30, 30, 0, 132, 90, 0, 270, 140, 0, 420, 270, 0, 600, 360, 0, 840, 330, 0, 1092, 660, 0, 1200, 810, 0, 1500, 570, 0, 1980, 1020, 0, 2190, 1260, 0, 2280, 1100, 0, 2460, 1560, 0, 3360, 1620, 0, 3780, 1452, 0, 3360, 2190, 0, 3930, 2340, 0, 4620, 1710, 0, 5400, 2940

Offset: 0

N. J. A. Sloane, Jan 31 2007

nonn

			1 + 30*q^4 + 30*q^6 + 132*q^10 + 90*q^12 + 270*q^16 + 140*q^18 + 420*q^22 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110 and 116.

John Cannon, Table of n, a(n) for n = 0..5000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices I: Quadratic Forms of Small Determinant, Proc. Royal Soc. London, Series A, 418 (1988), 17-41.
LMFDB, Integral lattice 5.162.12.1.1
G. Nebe and N. J. A. Sloane, Home page for this lattice

Cf. A008445, A125561, A023917.

al[n_, l_, p_, nn_] := Sum[Exp[-2 Pi I k l/n] EllipticTheta[3, Pi k/n, q^p]^n, {k, n}] / n / Sum[q^(p n (m + l/n)^2), {m, -nn, nn}] + O[q]^nn;
as[n_, s_, nn_] := CoefficientList[FullSimplify[Normal@Sum[al[n, l, n/s, nn], {l, s, n, s}]], q];
as[6, 1, 30] (*A023917*)
as[6, 2, 30][[;; ;; 2]] (*this sequence*)
as[6, 3, 30] (*A125561*)
(* Andrey Zabolotskiy, Feb 17 2022 *)

Typo in name corrected by Andrey Zabolotskiy, Feb 16 2022

A023918 Theta series of A*_6 lattice.

Original entry on oeis.org

1, 0, 0, 14, 0, 42, 70, 42, 0, 0, 210, 0, 294, 294, 210, 0, 0, 504, 0, 630, 882, 350, 0, 0, 1190, 0, 1470, 1148, 882, 0, 0, 1680, 0, 1708, 2520, 1050, 0, 0, 3150, 0, 3570, 2940, 1750, 0, 0, 3066, 0, 3864, 4774, 2100, 0, 0, 6174, 0, 5740, 5124, 3570, 0, 0, 6090

Offset: 0

N. J. A. Sloane

nonn

Positions of nonzero entries seem to be A047328. - Andrey Zabolotskiy, Nov 10 2021

			1 + 14*x^3 + 42*x^5 + 70*x^6 + 42*x^7 + 210*x^10 + 294*x^12 + 294*x^13 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.

S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_7(q).
G. Nebe and N. J. A. Sloane, Home page for this lattice

Cf. A008446.
Cf. theta series of lattices A*_0, A*_1, A*_2, A*_3, A*_4...: A000007, A000122, A004016, A004013, A023916, A023917, this sequence, A023919-A023936.
Cf. A047328.

a[n_] := Module[{A, A7}, A = x*O[x]^n; A7 = QPochhammer[x^7 + A]; A = QPochhammer[x + A]; SeriesCoefficient[A^7 / A7 + 7 * x * (A * A7)^3 + 7 * x^2 * A7^7 / A, {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from Michael Somos's PARI script *)

{a(n) = local(A, A7); if( n<0, 0, A = x * O(x^n); A7 = eta(x^7 + A); A = eta(x + A); polcoeff( A^7 / A7 + 7 * x * (A * A7)^3 + 7 * x^2 * A7^7 / A, n))}; /* Michael Somos, Jan 29 2011 */

Expansion of f(-x)^7 / f(-x^7) + 7 * x * f(-x)^3 * f(-x^7)^3 + 7 * x^2 * f(-x^7)^7 / f(-x) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 29 2011

a(7*n) = A008446(n). a(7*n + 1) = a(7*n + 2) = a(7*n + 4) = 0. - Michael Somos, Jan 29 2011

cp's OEIS Frontend

A072837 Duplicate of A023917.

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A072833 Numbers that are congruent to 0, 5, 8, 9 mod 12.

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A125564 Theta series of 5-dimensional lattice A_5^{+3}.

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A023918 Theta series of A*_6 lattice.

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