A089801 a(n) = 0 unless n = 3j^2 + 2j or 3j^2 + 4j + 1 for some j >= 0, in which case a(n) = 1.
1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
Offset: 0
Keywords
Examples
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 + ...
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.
- Index entries for characteristic functions.
Programs
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Magma
Basis( ModularForms( Gamma0(36), 1/2), 87) [2]; /* Michael Somos, Jul 02 2014 */
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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
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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 *) -
PARI
{a(n) = issquare(3*n + 1)}; /* Michael Somos, Apr 12 2005 */ -
Python
from sympy.ntheory.primetest import is_square def A089801(n): return int(is_square(3*n+1)) # Chai Wah Wu, May 17 2023
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
Extensions
Edited with simpler definition by N. J. A. Sloane, Jan 31 2012
Further edited by N. J. A. Sloane, Aug 12 2017
Comments