A144615 - OEIS

%I #30 Dec 17 2022 02:16:21

%S 3,6,15,12,24,18,42,24,42,30,63,48,60,42,84,48,93,54,120,60,96,84,126,

%T 72,114,96,186,84,132,90,168,120,171,102,210,108,216,114,210,144,186,

%U 156,255,132,204,138,336,168,222,150,300,192,240,192,294,168,324,174,372,180,336

%N a(n) = A000203(3n+2).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H Muniru A Asiru, <a href="/A144615/b144615.txt">Table of n, a(n) for n = 0..10000</a>

%F Expansion of q^(-2/3) * c(q)^2 / 3 in powers of q where c() is a cubic AGM theta function. - _Michael Somos_, Jun 07 2012

%F Expansion of q^(-2/3) * 3 * (eta(q^3)^3 / eta(q))^2 in powers of q. - _Michael Somos_, Jun 07 2012

%F a(n) = A000203(A016789(n)). - _Michel Marcus_, Jul 14 2015

%F a(n) = 3*A033686(n). - _Robert G. Wilson v_, Jan 12 2018

%F Sum_{k=1..n} a(k) = (2*Pi^2/9) * n^2 + O(n*log(n)). - _Amiram Eldar_, Dec 16 2022

%e G.f. = 3 + 6*x + 15*x^2 + 12*x^3 + 24*x^4 + 18*x^5 + 42*x^6 + 24*x^7 + 42*x^8 + ...

%e G.f. = 3*q^2 + 6*q^5 + 15*q^8 + 12*q^11 + 24*q^14 + 18*q^17 + 42*q^20 + 24*q^23 + ...

%p with(numtheory):

%p seq(sigma(3*n+2), n=0..10^3); # _Muniru A Asiru_, Dec 29 2017

%t a[ n_] := If[ n < 0, 0, DivisorSigma[ 1, 3 n + 2]]; (* _Michael Somos_, Jul 14 2015 *)

%t a[ n_] := SeriesCoefficient[ 3 (QPochhammer[ x^3]^3 / QPochhammer[ x])^2, {x, 0, n}]; (* _Michael Somos_, Jul 14 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * (eta(x^3 + A)^3 / eta(x + A))^2, n))}; /* _Michael Somos_, Jun 07 2012 */

%o (PARI) a(n)=sigma(3*n+2); \\ _Michel Marcus_, Jul 14 2015

%o (GAP) sequence := List([0..10^4],n->Sigma(3*n+2)); # _Muniru A Asiru_, Dec 29 2017

%o (Magma) [SumOfDivisors(3*n+2): n in [0..70]]; // _Vincenzo Librandi_, Jan 19 2018

%Y Cf. A000203, A016789, A033686.

%K nonn

%O 0,1

%A _N. J. A. Sloane_, Jan 15 2009