OFFSET
0,4
COMMENTS
Related to the Lorenz-Poincaré geometry of the group PSL[2,C]. - Roger L. Bagula, Feb 17 2006
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,4).
FORMULA
G.f.: (1-x)/(1 - 2*x + x^2 - 4*x^3).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, 2*k)*4^k.
MATHEMATICA
M = {{0, 1, 0}, {0, 0, 1}, {4, -1, 2}}; w[0] = {0, 1, 1}; w[n_] := w[n] = M.w[n - 1] a = Flatten[Table[w[n][[1]], {n, 0, 25}]] (* Roger L. Bagula, Feb 17 2006 *)
CoefficientList[Series[(1-x)/((1-x)^2-4x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, -1, 4}, {1, 1, 1}, 40] (* Harvey P. Dale, Jan 05 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1-x)/((1-x)^2-4*x^3)) \\ G. C. Greubel, Jun 06 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/((1-x)^2-4*x^3) )); // G. C. Greubel, Jun 06 2019
(Sage) ((1-x)/((1-x)^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019
(GAP) a:=[1, 1, 1];; for n in [4..30] do a[n]:=2*a[n-1]-a[n-2]+4*a[n-3]; od; a; # G. C. Greubel, Jun 06 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 25 2004
EXTENSIONS
Edited by N. J. A. Sloane, Aug 14 2008
Definition corrected by Harvey P. Dale, Jan 05 2019
STATUS
approved