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A089801
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G.f.: Sum_{n=-oo..oo} q^(3n^2+2n).
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7
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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, 0, 0, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Expansion of Jacobi theta function (theta_3(q^(1/3)) - theta_3(q^3))/(2 q^(1/3)).
Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| Cooper, S. and Hirschhorn, M. D., Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See X(q).
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.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
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FORMULA
| 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(3n + 1) where b(n) is multiplicative and 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
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(t) is 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
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EXAMPLE
| 1 + x + x^5 + x^8 + x^16 + x^21 + x^33 + x^40 + x^56 + x^65 + x^85 + ...
q + q^4 + q^16 + q^25 + q^49 + q^64 + q^100 + q^121 + q^169 + q^196 + ...
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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
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PROG
| (PARI) {a(n) = issquare(3*n + 1)} /* Michael Somos, Apr 12 2005 */
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CROSSREFS
| A089802(n) = (-1)^n * a(n). Characteristic function of A001082.
Sequence in context: A089802 A143064 A185124 A185125 A163811 A163817 A151667
Adjacent sequences: A089798 A089799 A089800 * A089802 A089803 A089804
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KEYWORD
| nonn,changed
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Nov 12 2003
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EXTENSIONS
| Edited with simpler definition by N. J. A. Sloane, Jan 31 2012
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