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

1,2

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).

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)

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 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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
(Sage) [lucas_number1(n, 8, 10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009
(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
(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

Crossrefs

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

Sequence in context: A081899 A057970 A208310 * A289796 A287814 A201640

Adjacent sequences: A154232 A154233 A154234 * A154236 A154237 A154238

Keyword

nonn,easy

Author

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

Extensions

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

Edited by Klaus Brockhaus, Oct 04 2009

Status

approved