A153596 a(n) = ((5 + sqrt(3))^n - (5 - sqrt(3))^n)/(2*sqrt(3)).

1, 10, 78, 560, 3884, 26520, 179752, 1214080, 8186256, 55152800, 371430368, 2500942080, 16837952704, 113358801280, 763153053312, 5137636904960, 34587001876736, 232842006858240, 1567506027294208, 10552536122060800, 71040228620135424

Offset

1,2

Comments

Third binomial transform of A054485. Fifth binomial transform of A162813 preceded by 1.

Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(3) = 6.73205080756887729....

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (10,-22).

Formula

G.f.: x/(1 - 10*x + 22*x^2). - Klaus Brockhaus, Dec 31 2008 [corrected Oct 11 2009]

a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009

E.g.f.: sinh(sqrt(3)*x)*exp(5*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016

Mathematica

Table[Simplify[((5+Sqrt[3])^n -(5-Sqrt[3])^n)/(2*Sqrt[3])], {n, 1, 25}] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011, modified by G. C. Greubel, Jun 01 2019 *)
LinearRecurrence[{10, -22}, {1, 10}, 25] (* G. C. Greubel, Aug 22 2016 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((5+r)^n-(5-r)^n)/(2*r): n in [1..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008
(Sage) [lucas_number1(n, 10, 22) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009
(Magma) I:=[1, 10]; [n le 2 select I[n] else 10*Self(n-1)-22*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016
(PARI) my(x='x+O('x^25)); Vec(x/(1-10*x+22*x^2)) \\ G. C. Greubel, Jun 01 2019

Crossrefs

Cf. A002194 (decimal expansion of sqrt(3)), A054485, A162813.

Sequence in context: A080618 A298270 A082136 * A316595 A348314 A056986

Adjacent sequences: A153593 A153594 A153595 * A153597 A153598 A153599

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

Extensions

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

Status

approved