OFFSET
0,4
COMMENTS
The first formula intersperses the terms with zeros, the second formula does not. - Colin Barker, Apr 02 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,-2,1,0,0,0,0,0,1,-2,1,0,-1,2,-1).
FORMULA
G.f.: (1 +x^8 +x^10 +x^12 +2*x^16 +2*x^20 +x^24 +x^26 +x^28 +x^36)/((1-x^2)*(1-x^6)*(1-x^8)*(1-x^24)), even terms only.
G.f.: (1 -x +x^3 +x^6 -x^7 +2*x^8 -x^9 +x^10 +x^13 -x^15 +x^16)/ ((1-x)^4* (1+x)^2*(1-x+x^2)*(1+x^2)^2*(1+x+x^2)*(1-x^2+x^4)). - Colin Barker, Apr 02 2015
EXAMPLE
G.f. = 1 + x^2 + x^4 + 2*x^6 + 4*x^8 + 5*x^10 + 7*x^12 + ...
MAPLE
S:=series( (1 +x^4 +x^5 +x^6 +2*x^8 +2*x^10 +x^12 +x^13 +x^14 +x^18 )/((1-x)*(1-x^3)*(1-x^4)*(1-x^12)), x, 65): seq(coeff(S, x, j), j=0..60); # G. C. Greubel, Feb 02 2020
MATHEMATICA
CoefficientList[Series[(1 +x^4 +x^5 +x^6 +2*x^8 +2*x^10 +x^12 +x^13 +x^14 +x^18 )/((1-x)*(1-x^3)*(1-x^4)*(1-x^12)), {x, 0, 60}], x] (* G. C. Greubel, Feb 02 2020 *)
PROG
(Magma) // Definition of group:
F<al> := CyclotomicField(24); i := al^6; eta := al^3; s2 := (1+i)/eta; om := al^8; s3 := (2*om+1)/i; s6 := s2*s3; M := GeneralLinearGroup(4, F);
A1 := M![s3/s6, 1/s6, 1/s6, 1/s6, -1/s6, s3/s6, -1/s6, 1/s6, -1/s6, 1/s6, s3/s6, -1/s6, -1/s6, -1/s6, 1/s6, s3/s6 ]; A2 := 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 ];
B1 := M![ -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1 ]; C1 := M![1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0 ];
C2 := M![0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0 ]; G := sub<M | A1, A2, B1, C1, C2 >;
(PARI) Vec((x^16-x^15+x^13+x^10-x^9+2*x^8-x^7+x^6+x^3-x+1) / ((x-1)^4*(x+1)^2*(x^2-x+1)*(x^2+1)^2*(x^2+x+1)*(x^4-x^2+1)) + O(x^60)) \\ Colin Barker, Apr 02 2015
(Sage)
def A078404_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1 +x^4 +x^5 +x^6 +2*x^8 +2*x^10 +x^12 +x^13 +x^14 +x^18 )/((1-x)*(1-x^3)*(1-x^4)*(1-x^12)) ).list()
A078404_list(60) # G. C. Greubel, Feb 02 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 27 2002
STATUS
approved