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Revision History for A033685

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A033685

Theta series of hexagonal lattice A_2 with respect to deep hole.
(history; published version)

#47 by Giovanni Resta at Thu Oct 13 05:58:04 EDT 2022

STATUS

reviewed

approved

#46 by Joerg Arndt at Thu Oct 13 04:53:18 EDT 2022

STATUS

proposed

reviewed

#45 by Amiram Eldar at Thu Oct 13 04:45:53 EDT 2022

STATUS

editing

proposed

#44 by Amiram Eldar at Thu Oct 13 04:39:19 EDT 2022

FORMULA

From Michael Somos, Dec 25 2011: (Start)

G.f.: Sum_{i, j in Z} x^(3 * (i^2 + i*j + j^2 + i + j) + 1). - _Michael Somos_, Dec 25 2011

G.f.: Sum_{i, j, k} x^(3 * Q(i, j, k) - 2) where Q(i, j, k) = i*i + j*j + k*k + i*j + i*k + j*k and the sum is over all integer i, j, k where i + j + k = 1. - _Michael Somos_, Dec 25 2011(End)

#43 by Amiram Eldar at Thu Oct 13 04:38:18 EDT 2022

LINKS

G. Gabriele Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>.

Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

#42 by Amiram Eldar at Thu Oct 13 04:37:35 EDT 2022

CROSSREFS

Cf. A000122, A000700, A010054, A121373.

#41 by Amiram Eldar at Thu Oct 13 04:36:32 EDT 2022

PROG

(PARI) {a(n) = if( (n<0) || (n%3 != 1), 0, 3 * sumdiv( n, d, kronecker( d, 3)))}; /* _\\ _Michael Somos_, Jul 16 2005 */

(PARI) {a(n) = my(A); if( (n<0) || (n%3 != 1), 0, n = n\3; A = x * O(x^n); 3 * polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; /* _\\ _Michael Somos_, Jul 16 2005 */

#40 by Amiram Eldar at Thu Oct 13 04:35:50 EDT 2022

FORMULA

From Amiram Eldar, Oct 13 2022: (Start)

a(n) = 3*A045833(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). (End)

CROSSREFS

Cf. A004016, A005882, A005928, A033687, A045833, A217219, A248897.

STATUS

approved

editing

#39 by Charles R Greathouse IV at Fri Mar 12 22:24:42 EST 2021

LINKS

M. Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

Discussion

Fri Mar 12

22:24

OEIS Server: https://oeis.org/edit/global/2897

#38 by N. J. A. Sloane at Wed Nov 13 21:58:46 EST 2019

LINKS

M. Somos, <a href="http://somos.crg4.comA010815/multiqa010815.htmltxt">Introduction to Ramanujan theta functions</a>

Discussion

Wed Nov 13

21:58

OEIS Server: https://oeis.org/edit/global/2832