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

Author

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

Keywords

• nonn

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).

Crossrefs

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

Programs

Magma

Z:= PolynomialRing(Integers()); N:=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

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

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 *)

PARI

my(x='x+O('x^25)); Vec(x/(1-10*x+22*x^2)) \\ G. C. Greubel, Jun 01 2019

Sage

[lucas_number1(n,10,22) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009

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

Extensions

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

Edited by Klaus Brockhaus, Oct 11 2009