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Numbers that are congruent to 0, 5, 8, 9 mod 12.
3

%I #39 Oct 27 2024 20:30:02

%S 0,5,8,9,12,17,20,21,24,29,32,33,36,41,44,45,48,53,56,57,60,65,68,69,

%T 72,77,80,81,84,89,92,93,96,101,104,105,108,113,116,117,120,125,128,

%U 129,132,137,140,141,144,149,152,153,156,161,164,165,168,173,176,177,180,185,188,189

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

%C 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

%H G. C. Greubel, <a href="/A072833/b072833.txt">Table of n, a(n) for n = 0..1000</a>

%H Scott Ahlgren, <a href="https://doi.org/10.1090/S0002-9939-99-05181-3">The sixth, eighth, ninth and tenth powers of Ramanujan's theta function</a>, Proc. Amer. Math. Soc. 128 (2000), 1333-1338.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F G.f.: x*(3*x^2-2*x+5) / ((x-1)^2*(x^2+1)). - _Colin Barker_, Jul 31 2013

%F 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

%F E.g.f.: exp(x)*(1 + 3*x) - cos(x) + sin(x). - _Stefano Spezia_, Oct 27 2024

%t 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 *)

%t Flatten[#+{0,5,8,9}&/@(12*Range[0,20])] (* _Harvey P. Dale_, Apr 10 2022 *)

%Y Cf. A023917, A108752, A072835.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jul 25 2002

%E Terms a(33) onward added by _G. C. Greubel_, Apr 16 2018

%E Edited by _Andrey Zabolotskiy_, Aug 14 2020