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

0,2

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).

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A244893.

Sequence in context: A075017 A060586 A076662 * A178307 A327465 A079063

Adjacent sequences: A164356 A164357 A164358 * A164360 A164361 A164362

Keyword

nonn,easy

Author

Michael Somos, Aug 13 2009

Status

approved