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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 (list; graph; refs; listen; history; text; internal format)
 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:=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 Adjacent sequences: A097114 A097115 A097116 * A097118 A097119 A097120 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

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Last modified September 29 23:18 EDT 2023. Contains 365781 sequences. (Running on oeis4.)