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
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).
Programs
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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
Comments