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 A015537 Expansion of x/(1 - 5*x - 4*x^2). 14
 0, 1, 5, 29, 165, 941, 5365, 30589, 174405, 994381, 5669525, 32325149, 184303845, 1050819821, 5991314485, 34159851709, 194764516485, 1110461989261, 6331368012245, 36098688018269, 205818912140325, 1173489312774701, 6690722212434805 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS First differences give A122690(n) = {1, 4, 24, 136, 776, 4424, 25224, ...}. Partial sums of a(n) are {0, 1, 6, 35, 200, ...} = (A123270(n) - 1)/8. - Alexander Adamchuk, Nov 03 2006 For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 5's along the main diagonal, and 2's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 19 2011 Pisano period lengths: 1, 1, 8, 1, 4, 8, 48, 1, 24, 4, 40, 8, 42, 48, 8, 2, 72, 24, 360, 4, ... - R. J. Mathar, Aug 10 2012 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Lucyna Trojnar-Spelina, Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7. Index entries for linear recurrences with constant coefficients, signature (5,4). FORMULA a(n) = 5*a(n-1) + 4*a(n-2). a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*4^k*5^(n-2*k-1). - Paul Barry, Apr 23 2005 a(n) = Sum_{k=0..(n-1)} A122690(k). - Alexander Adamchuk, Nov 03 2006 a(n) = (1/41)*sqrt(41)*((5/2 + (1/2)*sqrt(41))^n - (5/2 - (1/2)*sqrt(41))^n), with n >= 0. - Paolo P. Lava, Jan 13 2009 a(n) = 2^(n-1)*Fibonacci(n, 5/2) = (2/i)^(n-1)*ChebyshevU(n-1, 5*i/4). - G. C. Greubel, Dec 26 2019 MAPLE seq( simplify((2/I)^(n-1)*ChebyshevU(n-1, 5*I/4)), n=0..20); # G. C. Greubel, Dec 26 2019 MATHEMATICA LinearRecurrence[{5, 4}, {0, 1}, 30] (* Vincenzo Librandi, Nov 12 2012 *) Table[2^(n-1)*Fibonacci[n, 5/2], {n, 0, 30}] (* G. C. Greubel, Dec 26 2019 *) PROG (Sage) [lucas_number1(n, 5, -4) for n in range(0, 22)] # Zerinvary Lajos, Apr 24 2009 (Magma) [n le 2 select n-1 else 5*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 12 2012 (PARI) x='x+O('x^30); concat([0], Vec(x/(1-5*x-4*x^2))) \\ G. C. Greubel, Jan 01 2018 (GAP) a:=[0, 1];; for n in [3..30] do a[n]:=5*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019 CROSSREFS Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015443, A015447, A030195, A053404, A057087, A083858, A085939, A090017, A091914, A099012, A122690, A123270, A180222, A180226. Sequence in context: A272940 A146178 A272751 * A182017 A291017 A141812 Adjacent sequences: A015534 A015535 A015536 * A015538 A015539 A015540 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified December 6 03:23 EST 2022. Contains 358594 sequences. (Running on oeis4.)