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A154236
a(n) = ( (5 + sqrt(6))^n - (5 - sqrt(6))^n )/(2*sqrt(6)).
2
1, 10, 81, 620, 4661, 34830, 259741, 1935640, 14421321, 107436050, 800355401, 5962269060, 44415937981, 330876267670, 2464859855061, 18361949464880, 136787157402641, 1018994534193690, 7590989351286721, 56548997363187100
OFFSET
1,2
COMMENTS
First differences are in A164551.
Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(6) = 7.4494897427....
FORMULA
From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 10*a(n-1) - 19*a(n-2) for n > 1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 10*x + 19*x^2). (End)
MATHEMATICA
LinearRecurrence[{10, -19}, {1, 10}, 25] (* or *) Table[Simplify[((5 + Sqrt[6])^n -(5-Sqrt[6])^n)/(2*Sqrt[6])], {n, 1, 25}] (* G. C. Greubel, Sep 06 2016 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-6); S:=[ ((5+r)^n-(5-r)^n)/(2*r): n in [1..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009
(Sage) [lucas_number1(n, 10, 19) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009
(PARI) a(n)=([0, 1; -19, 10]^(n-1)*[1; 10])[1, 1] \\ Charles R Greathouse IV, Sep 07 2016
CROSSREFS
Cf. A010464 (decimal expansion of square root of 6), A164551.
Sequence in context: A136870 A293367 A018202 * A095004 A037541 A037485
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