A164359 Expansion of (1 - x^2)^3 / ((1 - x)^3 * (1 - x^3)) in powers of x.

1, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3

Offset: 0

Author

Michael Somos, Aug 13 2009

Keywords

• nonn
• easy

Examples

			G.f. = 1 + 3*x + 3*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 2*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Crossrefs

Cf. A244893.

Programs

Mathematica

a[ n_] := -Boole[n == 0] + 2 + KroneckerSymbol[ 9, n]; (* Michael Somos, Apr 17 2015 *)
CoefficientList[Series[(1-x^2)^3/((1-x)^3*(1-x^3)),{x,0,120}],x] (* or *) LinearRecurrence[{0,0,1},{1,3,3,2},120] (* or *) PadRight[{1},120,{2,3,3}] (* Harvey P. Dale, Aug 16 2021 *)

PARI

{a(n) = -(n==0) + 2 + kronecker(9, n)};

Formula

Euler transform of length 3 sequence [ 3, -3, 1].

Moebius transform is length 3 sequence [ 3, 0, -1].

a(-n) = a(n) for all n in Z. a(n+3) = a(n) unless n=0 or n=-3. a(3*n) = 2 unless n=0. a(3*n + 1) = a(3*n + 2) = 3.

G.f.: -1 + (1/3) * ( 8 / (1 - x) - (2 + x) / (1 + x + x^2) ).

a(n) = A244893(n) if n>1. - Michael Somos, Apr 17 2015