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

Author

N. J. A. Sloane, Jul 25 2002

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A023917, A108752, A072835.

Programs

Mathematica

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 *)

Formula

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

Extensions

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

Edited by Andrey Zabolotskiy, Aug 14 2020