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 A015530 Expansion of x/(1 - 4*x - 3*x^2). 23
 0, 1, 4, 19, 88, 409, 1900, 8827, 41008, 190513, 885076, 4111843, 19102600, 88745929, 412291516, 1915403851, 8898489952, 41340171361, 192056155300, 892245135283, 4145149007032, 19257331433977, 89464772757004 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Let b(1)=1, b(k) = floor(b(k-1)) + 3/b(k-1); then for n>1, b(n) = a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002 In general, x/(1 - a*x - b*x^2) has a(n) = sum_{k=0..floor((n-1)/2)} C(n-k-1,k)*b^k*a^(n-2k-1). - Paul Barry, Apr 23 2005 Pisano period lengths: 1, 2, 1, 4, 24, 2, 21, 4, 3, 24, 40, 4, 84, 42, 24, 8, 288, 6, 18, 24, ... . - R. J. Mathar, Aug 10 2012 This is the Lucas sequence U(4,-3). - Bruno Berselli, Jan 09 2013 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. Wikipedia, Lucas sequence. Index entries for linear recurrences with constant coefficients, signature (4,3). FORMULA a(n) = 4*a(n-1) + 3*a(n-2). a(n) = (A086901(n+2) - A086901(n+1))/6. - Ralf Stephan, Feb 01 2004 a(n) = sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*3^k*4^(n-2k-1). - Paul Barry, Apr 23 2005 a(n) = ((2+sqrt(7))^n - (2-sqrt(7))^n)/sqrt(28). Offset 1. a(3)=19. - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009 From Johannes W. Meijer, Aug 01 2010: (Start) Limit(a(n+k)/a(k), k=infinity) = A108851(n)+a(n)*sqrt(7) Limit(A108851(n)/a(n), n=infinity) = sqrt(7) (End) G.f.: x*G(0) where G(k)= 1 + (4*x+3*x^2)/(1 - (4*x+3*x^2)/(4*x + 3*x^2 + 1/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 28 2012 G.f.: G(0)*x/(2-4*x), where G(k)= 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013 MATHEMATICA a[n_]:=(MatrixPower[{{1, 6}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *) LinearRecurrence[{4, 3}, {0, 1}, 30] (* Vincenzo Librandi, Jun 19 2012 *) PROG (Sage) [lucas_number1(n, 4, -3) for n in range(0, 23)]# Zerinvary Lajos, Apr 23 2009 (MAGMA) I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 19 2012 (PARI) x='x+O('x^30); concat([0], Vec(x/(1-4*x-3*x^2))) \\ G. C. Greubel, Jan 24 2018 CROSSREFS Appears in A179596, A126473 and A179597. - Johannes W. Meijer, Aug 01 2010 Cf. A080042: Lucas sequence V(4,-3). Sequence in context: A291416 A192526 A084155 * A256959 A181880 A291016 Adjacent sequences:  A015527 A015528 A015529 * A015531 A015532 A015533 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified May 7 20:36 EDT 2021. Contains 343652 sequences. (Running on oeis4.)