A014018 -id:A014018 - cp's OEIS frontend

A025765 Expansion of 1/((1-x)(1-x^2)(1-x^9)).

1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 49, 51, 54, 56, 59, 61, 64, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 100, 103, 107, 110

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,0,0,0,1,-1,-1,1).

A014018 := proc(n) if n < 0 then 0; else coeftayl(1/(1+x^3+x^6),x=0,n) ; end if; end proc:
A061347 := proc(n) op(1+(n mod 3),[1,1,-2]) ; end proc:
A025765 := proc(n) 1/3*n +173/216 +1/36*n^2 +1/8*(-1)^n + ( A014018(n-2)+A014018(n-4)+A014018(n-5))/3 - A061347(n+2)/27 ; end proc:
seq(A025765(n),n=0..40) ; # R. J. Mathar, Mar 22 2011

CoefficientList[Series[1/((1-x)(1-x^2)(1-x^9)),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,-1,0,0,0,0,0,1,-1,-1,1},{1,1,2,2,3,3,4,4,5,6,7,8},60] (* Harvey P. Dale, Aug 14 2021 *)

Vec(1/((1-x)*(1-x^2)*(1-x^9))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n)= +a(n-1) +a(n-2) -a(n-3) +a(n-9) -a(n-10) -a(n-11) +a(n-12). - R. J. Mathar, Mar 22 2011

A099917 Expansion of (1+x^2)^2/(1+x^3+x^6).

Original entry on oeis.org

1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0, 2, -1, 1, -2, 0, -1, 0, 1, 0

Offset: 0

Paul Barry, Oct 30 2004

The denominator is the 9th cyclotomic polynomial. The g.f. is a Chebyshev transform of that of (-1)^n*A052931(n) by the Chebyshev mapping g(x)->(1/(1+x^2))g(x/(1+x^2)). The reciprocal of the 9th cyclotomic polynomial A014018 is given by sum{k=0..n, A099917(n-k)(k/2+1)(-1)^(k/2)(1+(-1)^k)/2}.

Index entries for linear recurrences with constant coefficients, signature (0, 0, -1, 0, 0, -1).

Cf. A099916.

a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*sum{j=0..n-2k, C(j, n-2k-2j)3^k(-1/3)^(n-2k)}}; a(n)=sum{k=0..n, A014018(n-k)C(2, k/2)(1+(-1)^k)/2}.

cp's OEIS Frontend

A025765 Expansion of 1/((1-x)(1-x^2)(1-x^9)).

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