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A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3. 51
1, 1, 2, 0, 2, 1, 2, 0, 1, 2, 2, 0, 2, 0, 2, 0, 3, 2, 0, 0, 2, 1, 2, 0, 2, 2, 2, 0, 0, 0, 4, 0, 2, 1, 2, 0, 2, 2, 0, 0, 1, 2, 2, 0, 4, 0, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 3, 2, 2, 0, 2, 0, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 2, 4, 0, 0, 1, 4, 0, 0, 2, 2, 0, 2, 0, 2, 0, 1, 2, 0, 0, 4, 2, 2, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
a(n)=0 if and only if A000731(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Number of 3-core partitions of n (denoted c_3(n) in Granville and Ono, p. 340). - Brian Hopkins, May 13 2008
Denoted by g_1(q) in Cynk and Hulek in Remark 3.4 on page 12 (but not explicitly listed).
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.35) and (32.351).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 697.
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
G.-N. Han and Ken Ono, Hook lengths and 3-cores
J. Lovejoy and O. Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci., 6 (2008), 23-36. [From Jeremy Lovejoy, Jun 12 2009]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 3 sequence [1, 1, -2, ...].
Expansion of q^(-1/3) * eta(q^3)^3 / eta(q) in powers of q.
a(4*n + 1) = a(n). - Michael Somos, Dec 06 2004
a(n) = b(3*n + 1) where b(n) is multiplicative and b(p^e) = 0^e if p = 3, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6). - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3. - Michael Somos, Dec 06 2004
Given g.f. A(x), B(q)= q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3. - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. - Michael Somos, May 20 2005
G.f.: Product_{k>0} (1 - x^(3*k))^3 / (1 - x^k).
G.f.: Sum_{k in Z} x^k / (1 - x^(3*k + 1)) = Sum_{k in Z} x^k / (1 - x^(6*k + 2)). - Michael Somos, Nov 03 2005
Expansion of q^(-1) * c(q^3) / 3 = q^(-1) * (a(q) - b(q)) / 9 in powers of q^3 where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A005928.
a(n) = Sum_{d|3n+1} LegendreSymbol{d,3} - Brian Hopkins, May 13 2008
q-series for a(n): Sum_{n >= 0} q^(n^2+n)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))). [From Jeremy Lovejoy, Jun 12 2009]
a(n) = A002324(3*n + 1). 3*a(n) = A005882(n) = A033685(3*n + 1). - Michael Somos, Apr 04 2003
G.f.: (2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5)) / 3 where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 07 2018
Sum_{k=1..n} a(k) ~ 2*Pi*n/3^(3/2). - Vaclav Kotesovec, Dec 17 2022
EXAMPLE
G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + 2*x^12 + 2*x^14 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + 2*q^37 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Sep 23 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Sep 01 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, sumdiv( 3*n + 1, d, kronecker( -3, d)))}; /* Michael Somos, Nov 03 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / eta(x + A), n))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor( 3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%6==1, e+1, 1-e%2)))}; /* Michael Somos, May 06 2015 */
(Magma) Basis( ModularForms( Gamma1(9), 1), 316) [2]; /* Michael Somos, May 06 2015 */
CROSSREFS
Sequence in context: A068907 A219762 A227696 * A263452 A133457 A324120
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified April 25 03:15 EDT 2024. Contains 371964 sequences. (Running on oeis4.)