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A089801 a(n) = 0 unless n = 3j^2 + 2j or 3j^2 + 4j + 1 for some j >= 0, in which case a(n) = 1.

%I A089801 #96 Feb 16 2025 08:32:51
%S A089801 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,
%T A089801 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,
%U A089801 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
%N A089801 a(n) = 0 unless n = 3j^2 + 2j or 3j^2 + 4j + 1 for some j >= 0, in which case a(n) = 1.
%C A089801 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A089801 Also characteristic function of generalized octagonal numbers A001082. - _Omar E. Pol_, Jul 13 2012
%C A089801 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
%H A089801 Seiichi Manyama, <a href="/A089801/b089801.txt">Table of n, a(n) for n = 0..1000</a>
%H A089801 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/320.pdf">The Bhargava-Adiga Summation and Partitions</a>, 2016. See Th. 2.
%H A089801 S. Cooper and M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00079-7">Results of Hurwitz type for three squares.</a> Discrete Math. 274 (2004), no. 1-3, 9-24. See X(q).
%H A089801 James A. Sellers, <a href="https://arxiv.org/abs/2409.12321">Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice</a>, arXiv:2409.12321 [math.NT], 2024. See p. 4.
%H A089801 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A089801 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%H A089801 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>.
%H A089801 I. J. Zucker, <a href="http://dx.doi.org/10.1088/0305-4470/23/2/009">Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums</a>, J. Phys. A: Math. Gen. 23, 117-132, 1990.
%H A089801 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F A089801 G.f.: Sum_{n=-oo..oo} q^(3n^2+2n).
%F A089801 Expansion of Jacobi theta function (theta_3(q^(1/3)) - theta_3(q^3))/(2 q^(1/3)) in powers of q.
%F A089801 Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, -1, ...]. - _Michael Somos_, Apr 12 2005
%F A089801 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
%F A089801 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
%F A089801 Expansion of chi(x) * psi(-x^3) in powers of x where chi(), psi() are Ramanujan theta functions. - _Michael Somos_, Apr 19 2007
%F A089801 Expansion of f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Jun 29 2012
%F A089801 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.
%F A089801 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
%F A089801 a(n) = (-1)^n * A089802(n).
%F A089801 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
%F A089801 a(n) = A033684(3*n + 1). - _Michael Somos_, Jan 10 2017
%e A089801 G.f. = 1 + x + x^5 + x^8 + x^16 + x^21 + x^33 + x^40 + x^56 + x^65 + x^85 + ...
%e A089801 G.f. = q + q^4 + q^16 + q^25 + q^49 + q^64 + q^100 + q^121 + q^169 + q^196 + ...
%p A089801 A089801 := proc(n)
%p A089801         A033684(3*n+1) ;
%p A089801 end proc: # _R. J. Mathar_, Oct 07 2011
%p A089801 M:=33;
%p A089801 S:=f->series(f,q,500);
%p A089801 L:=f->seriestolist(f);
%p A089801 X:=add(q^(3*n^2+2*n),n=-M..M);
%p A089801 S(%);
%p A089801 L(%); # _N. J. A. Sloane_, Jan 31 2012
%p A089801 eps:=Array(0..120,0);
%p A089801 for j from 0 to 120 do
%p A089801 if 3*j^2+2*j <= 120 then eps[3*j^2+2*j] := 1; fi;
%p A089801 if 3*j^2+4*j+1 <= 120 then eps[3*j^2+4*j+1] := 1; fi;
%p A089801 end do;  # _N. J. A. Sloane_, Aug 12 2017
%t A089801 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 *)
%t A089801 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 *)
%o A089801 (PARI) {a(n) = issquare(3*n + 1)}; /* _Michael Somos_, Apr 12 2005 */
%o A089801 (Magma) Basis( ModularForms( Gamma0(36), 1/2), 87) [2]; /* _Michael Somos_, Jul 02 2014 */
%o A089801 (Python)
%o A089801 from sympy.ntheory.primetest import is_square
%o A089801 def A089801(n): return int(is_square(3*n+1)) # _Chai Wah Wu_, May 17 2023
%Y A089801 Cf. A001082, A033684, A080995, A089802, A089807, A280739.
%K A089801 nonn
%O A089801 0,1
%A A089801 _Eric W. Weisstein_, Nov 12 2003
%E A089801 Edited with simpler definition by _N. J. A. Sloane_, Jan 31 2012
%E A089801 Further edited by _N. J. A. Sloane_, Aug 12 2017