0,3

Case k=4, i=3 of Gordon Theorem.

Expansion of q^(-1/12)eta(q^3)/eta(q) in powers of q. - Michael Somos Apr 20 2004

Euler transform of period 3 sequence [1,1,0,...]. - Michael Somos Apr 20 2004

Also number of partitions with at most 2 parts of size 1 and all differences between parts at distance 3 are greater than 1. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,2] (for example, [2,2,1,1] does not qualify because the difference between the first and the fourth parts is equal to 1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006

Also number of partitions of n where no positive integer appears more than twice. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006

Also number of partitions of n with least part either 1 or 2 and with differences of consecutive parts at most 2. Example: a(6)=7 because we have [4,2],[3,2,1],[3,1,1,1],[2,2,2],[2,2,1,1],[2,1,1,1,1] and [1,1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145.

T. D. Noe, Table of n, a(n) for n=0..1000

N. Chair, Partition identities from Partial Supersymmetry

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Given g.f. A(x) then B(x)=x*A(x^6)^2 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u,v,w)= +v^2 +v*w^2 -v*u^2 +3*u^2*w^2 . - Michael Somos May 28 2006

G.f.: 1/(Product_{k>0} (1-x^(3k-1))(1-x^(3k-2))) = Product_{k>0} 1+x^k+x^(2k) (where 1+x+x^2 is 3rd cyclotomic polynomial).

a(6)=7 because we have [5,1],[4,2],[4,1,1],[2,2,2],[2,2,1,1],[2,1,1,1,1] and [1,1,1,1,1,1].

g:=product(1+x^j+x^(2*j), j=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..50); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006

f[0] = 1; f[n_] := Coefficient[Expand@ Product[1 + x^k + x^(2k), {k, n}], x^n]; Table[f@n, {n, 0, 40}] (* from Robert G. Wilson v (rgwv(at)rgwv.com), Nov 10 2006 *)

(PARI) a(n)=if(n<0, 0, polcoeff(eta(x^3+x*O(x^n))/eta(x+x*O(x^n)), n))

Cf. A001935, A035959. a(n)=A061197(n, n).

Cf. A001651, A003105, A035361, A035360.

Sequence in context: A166239 A058661 A094362 this_sequence A128663 A135833 A137200

Adjacent sequences: A000723 A000724 A000725 this_sequence A000727 A000728 A000729

nonn,easy,nice

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

More terms from Olivier Gerard (olivier.gerard(AT)gmail.com)