0,6

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).

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

Has a nice Dirichlet series expansion, see PARI line.

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

G.f.: Product_{k>0} (1+x^k)(1-x^(3k))(1-x^(8k))/(1+x^(12k)).

G.f.: 1 + Sum_{k>0} x^k(1+x^(4k))(1+x^(6k))/(1+x^(12k)) . - Michael Somos Sep 10 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

Expansion of (phi(q)phi(q^6)+phi(q^2)phi(q^3))/2 where phi() is a Ramanujan theta function. - 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.

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.: 1 + Sum{n = -infinity...infinity} (q^n + q^(5n)) / (1 + q^(12n)) (see Berkovich/Yesilyurt). - Ralf Stephan, May 14 2007

(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))

Sequence in context: A029439 A075117 A029810 this_sequence A115660 A128581 A026517

Adjacent sequences: A000374 A000375 A000376 this_sequence A000378 A000379 A000380

nonn,easy,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).

Edited by Michael Somos, Sep 10, 2002.