OFFSET
0,1
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also characteristic function of generalized octagonal numbers A001082. - Omar E. Pol, Jul 13 2012
Number 12 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Th. 2.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See X(q).
James A. Sellers, Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice, arXiv:2409.12321 [math.NT], 2024. See p. 4.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.
FORMULA
G.f.: Sum_{n=-oo..oo} q^(3n^2+2n).
Expansion of Jacobi theta function (theta_3(q^(1/3)) - theta_3(q^3))/(2 q^(1/3)) in powers of q.
Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, -1, ...]. - Michael Somos, Apr 12 2005
a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p != 3. - Michael Somos, Jun 06 2005; b=A033684. - R. J. Mathar, Oct 07 2011
Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q. - Michael Somos, Apr 12 2005
Expansion of chi(x) * psi(-x^3) in powers of x where chi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 19 2007
Expansion of f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 29 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089807.
a(8*n + 4) = a(4*n + 2) = a(4*n + 3) = 0, a(4*n + 1) = a(n), a(8*n) = A080995(n). - Michael Somos, Jan 28 2011
a(n) = (-1)^n * A089802(n).
For n > 0, a(n) = b(n)-b(n-1) + c(n)-c(n-1), where b(n) = floor(sqrt(n/3+1/9)+2/3) and c(n) = floor(sqrt(n/3+1/9)+4/3). - Mikael Aaltonen, Jan 22 2015
a(n) = A033684(3*n + 1). - Michael Somos, Jan 10 2017
EXAMPLE
G.f. = 1 + x + x^5 + x^8 + x^16 + x^21 + x^33 + x^40 + x^56 + x^65 + x^85 + ...
G.f. = q + q^4 + q^16 + q^25 + q^49 + q^64 + q^100 + q^121 + q^169 + q^196 + ...
MAPLE
A089801 := proc(n)
A033684(3*n+1) ;
end proc: # R. J. Mathar, Oct 07 2011
M:=33;
S:=f->series(f, q, 500);
L:=f->seriestolist(f);
X:=add(q^(3*n^2+2*n), n=-M..M);
S(%);
L(%); # N. J. A. Sloane, Jan 31 2012
eps:=Array(0..120, 0);
for j from 0 to 120 do
if 3*j^2+2*j <= 120 then eps[3*j^2+2*j] := 1; fi;
if 3*j^2+4*j+1 <= 120 then eps[3*j^2+4*j+1] := 1; fi;
end do; # N. J. A. Sloane, Aug 12 2017
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/3) (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3]), {x, 0, n}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Jun 29 2012 *)
PROG
(PARI) {a(n) = issquare(3*n + 1)}; /* Michael Somos, Apr 12 2005 */
(Magma) Basis( ModularForms( Gamma0(36), 1/2), 87) [2]; /* Michael Somos, Jul 02 2014 */
(Python)
from sympy.ntheory.primetest import is_square
def A089801(n): return int(is_square(3*n+1)) # Chai Wah Wu, May 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Nov 12 2003
EXTENSIONS
Edited with simpler definition by N. J. A. Sloane, Jan 31 2012
Further edited by N. J. A. Sloane, Aug 12 2017
STATUS
approved