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A000377
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Sum over divisors d of n of Kronecker symbol (-6, d), with a(0) = 1.
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6
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1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 0, 2, 2, 1, 0, 1, 0, 2, 2, 2, 0, 1, 3, 0, 1, 2, 2, 2, 2, 1, 2, 0, 4, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 1, 3, 3, 0, 0, 2, 1, 4, 2, 0, 2, 2, 2, 0, 2, 2, 1, 0, 2, 0, 0, 0, 4, 0, 1, 2, 0, 3, 0, 4, 0, 2, 2, 1, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 1, 2, 3, 2, 3, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| G. E. Andrews, editor, P. A. MacMahon: Collected Papers Volume II, MIT Press, 1986, p. 260.
G. E. Andrews, "Nathan Fine 1916-1994", Notices Amer. Math. Soc., 42 (No. 6, 1995), 678-679.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 81, Eq. (32.5).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Michael Gilleland, Some Self-Similar Integer Sequences
Eric Weisstein's World of Mathematics, Fine's Equation
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms
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FORMULA
| Expansion of (phi(q) * phi(q^6) + phi(q^2) * phi(q^3)) / 2 = psi(-q^2) * psi(-q^3) * chi(-q^6) * chi(-q^12) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos Jan 26 2006
Expansion of eta(q^2) * eta(q^3) * eta(q^8) * eta(q^12) / (eta(q) * eta(q^24)) in powers of q.
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), a(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24). - Michael Somos Jun 17 2005
Moebius transform is period 24 sequence [ 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, ...]. - Michael Somos Jan 26 2006
Euler transform of period 24 sequence [ 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24*t)) = 24^(1/2)*(t/i)*f(t) where q = exp(2*pi*i*t). - Michael Somos Jun 22 2011
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1-x^(8*k)) / (1 + x^(12*k)).
G.f.: 1 + Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)). - Michael Somos Sep 10 2005
G.f.: 1 + Sum{n = -infinity...infinity} (q^n + q^(5*n)) / (1 + q^(12*n)) (see Berkovich/Yesilyurt). - Ralf Stephan, May 14 2007
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EXAMPLE
| 1 + q + q^2 + q^3 + q^4 + 2*q^5 + q^6 + 2*q^7 + q^8 + q^9 + 2*q^10 + ...
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MATHEMATICA
| a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -6, #] &]] (* Michael Somos Jul 11 2011 *)
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PROG
| (PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -6, d)))}
(PARI) {a(n) = if( n<1, n==0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -6, p) * X))[n])}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^24 + A)), n))}
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CROSSREFS
| Sequence in context: A029439 A075117 A029810 * A190611 A115660 A128581
Adjacent sequences: A000374 A000375 A000376 * A000378 A000379 A000380
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KEYWORD
| nonn,easy,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Michael Somos, Sep 10, 2002.
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