A164358 Expansion of (1 - x^2)^2 * (1 - x^3) / ((1 - x)^3 * (1 - x^4)) in powers of x.

1, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2

Offset: 0

Author

Michael Somos, Aug 13 2009

Keywords

• nonn
• easy

Comments

Coordination sequence for a chain of hexagons joined by single edges. - N. J. A. Sloane, Nov 21 2019

Examples

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

Links

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

Programs

Magma

m:=150; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x+4*x^2+3*x^3+x^4)/(1-x^4))); // G. C. Greubel, Sep 26 2018

Mathematica

a[ n_] := - Boole[n == 0] + 3 - If[ EvenQ[n], (-1)^(n/2), 0];
CoefficientList[Series[(1+3*x+4*x^2+3*x^3+x^4)/(1-x^4), {x, 0, 150}], x] (* G. C. Greubel, Sep 26 2018 *)

PARI

{a(n) = -(n==0) + 3 - if( n%2 == 0, (-1)^(n/2), 0)};

PARI

x='x+O('x^150); Vec((1+3*x+4*x^2+3*x^3+x^4)/(1-x^4)) \\ G. C. Greubel, Sep 26 2018

Formula

a(n) = 3*b(n) unless n=0 where b(n) is multiplicative with b(2) = 4/3, b(2^e) = 2/3 if e>1, b(p^e) = 1 if p>2.

Euler transform of length 4 sequence [3, -2, -1, 1].

Moebius transform is length 4 sequence [3, 1, 0, -2].

a(n) = a(-n) for all n in Z. a(n+4) = a(n) unless n=0 or n=-4. a(4*n) == 2 unless n=0. a(2*n + 1) = 3. a(4*n + 2) = 4.

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

G.f.: (1+x)*(1+x+x^2)/((1-x)*(1+x^2)). - N. J. A. Sloane, Nov 21 2019