A154235 a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).

1, 8, 54, 352, 2276, 14688, 94744, 611072, 3941136, 25418368, 163935584, 1057300992, 6819052096, 43979406848, 283644733824, 1829363802112, 11798463078656, 76094066608128, 490767902078464, 3165202550546432

Offset: 1

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Keywords

• nonn
• easy

Comments

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(6) = 6.4494897427....

Binomial transform of A164550, second binomial transform of A164549, third binomial transform of A123011, fourth binomial transform of A164532.

Binomial transform is A164551, second binomial transform is A164552, third binomial transform is A164553.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-10).

Crossrefs

Cf. A010464 (decimal expansion of square root of 6), A123011, A164532, A164549, A164550, A164551, A164552, A164553.

Programs

GAP

a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-10*a[n-2]; od; a; # G. C. Greubel, May 21 2019

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

Mathematica

LinearRecurrence[{8, -10}, {1, 8}, 30] (* or *) Table[Simplify[((4 + Sqrt[6])^n -(4-Sqrt[6])^n)/(2*Sqrt[6])], {n, 30}] (* G. C. Greubel, Sep 06 2016 *)

PARI

a(n)=([0,1; -10,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

PARI

my(x='x+O('x^30)); Vec(x/(1-8*x+10*x^2)) \\ G. C. Greubel, May 21 2019

Sage

[lucas_number1(n,8,10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

Formula

From Philippe Deléham, Jan 06 2009: (Start)

a(n) = 8*a(n-1) - 10*a(n-2) for n > 1, where a(0)=0, a(1)=1.

G.f.: x/(1 - 8*x + 10*x^2). (End)

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 04 2009