OFFSET
0,3
COMMENTS
The first formula intersperses the terms with zeros, the second formula doesn't. - Colin Barker, Apr 01 2015
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,2,-2,1,0,-1,2,-1).
FORMULA
G.f.: (x^22+x^16+x^14+x^12+x^10+x^8+x^6+1)/((1-x^2)*(1-x^4)*(1-x^8)*(1-x^12)).
G.f.: (x^10-x^9+x^8+x^6+x^4+x^2-x+1) / ((x-1)^4*(x+1)^2*(x^2-x+1)*(x^2+1)*(x^2+x+1)). - Colin Barker, Apr 01 2015
EXAMPLE
1 + x^2 + 2*x^4 + 3*x^6 + 6*x^8 + 8*x^10 + 13*x^12 + 17*x^14 + 25*x^16 + 31*x^18 + ...
MATHEMATICA
LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 1, 0, -1, 2, -1}, {1, 1, 2, 3, 6, 8, 13, 17, 25, 31, 42, 52}, 60] (* Harvey P. Dale, Aug 29 2016 *)
PROG
(Magma) // Definition of group: F<al> := CyclotomicField(12); w := al^4; i := al^3; s3 := (1+2*w)/i; M := GeneralLinearGroup(4, F);
g1 := M![ 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0 ]; g2 := M![ -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0 ]; g3 := M![ 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1 ];
H := M![ 0, -1/s3, -1/s3, -1/s3, 1/s3, 0, 1/s3, -1/s3, 1/s3, -1/s3, 0, 1/s3, 1/s3, 1/s3, -1/s3, 0 ]; G := sub<M| g1, g2, g3, H>;
(PARI) Vec((x^10-x^9+x^8+x^6+x^4+x^2-x+1) / ((x-1)^4*(x+1)^2*(x^2-x+1)*(x^2+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Apr 01 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 31 2002
STATUS
approved