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A097117
Expansion of (1-x)/((1-x)^2 - 4*x^3).
3
1, 1, 1, 5, 13, 25, 57, 141, 325, 737, 1713, 3989, 9213, 21289, 49321, 114205, 264245, 611569, 1415713, 3276837, 7584237, 17554489, 40632089, 94046637, 217679141, 503840001, 1166187409, 2699251381, 6247675357, 14460848969, 33471028105
OFFSET
0,4
COMMENTS
Related to the Lorenz-Poincaré geometry of the group PSL[2,C]. - Roger L. Bagula, Feb 17 2006
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
Sequence in context: A241657 A249513 A147090 * A146140 A146283 A026373
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