A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1.

First differences are in A161728.

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729....

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (8,-13).

Crossrefs

Cf. A002194 (decimal expansion of sqrt(3)), A054491, A074324, A161728, A162766.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ];  // Klaus Brockhaus, Dec 31 2008

Magma

I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

Mathematica

Join[{a=1,b=8},Table[c=8*b-13*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8,-13},{1,8},40] (* Harvey P. Dale, Aug 16 2012 *)

PARI

a(n)=([0,1; -13,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 04 2016

Sage

[lucas_number1(n,8,13) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

Formula

G.f.: x/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009

a(n) = 8*a(n-1) - 13*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(4*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016

a(n) = Sum_{k=0..n-1} A027907(n,2k+1)*3^k. - J. Conrad, Aug 30 2016

a(n) = Sum_{k=0..n-1} A083882(n-1-k)*4^k. - J. Conrad, Sep 03 2016

Extensions

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

Edited by Klaus Brockhaus, Oct 11 2009