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A154235
a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).
5
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.
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
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