A154236 a(n) = ( (5 + sqrt(6))^n - (5 - sqrt(6))^n )/(2*sqrt(6)).

1, 10, 81, 620, 4661, 34830, 259741, 1935640, 14421321, 107436050, 800355401, 5962269060, 44415937981, 330876267670, 2464859855061, 18361949464880, 136787157402641, 1018994534193690, 7590989351286721, 56548997363187100

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

First differences are in A164551.

Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(6) = 7.4494897427....

Links

• G. C. Greubel, Table of n, a(n) for n = 1..999
• Index entries for linear recurrences with constant coefficients, signature (10, -19).

Crossrefs

Cf. A010464 (decimal expansion of square root of 6), A164551.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=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

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 *)

PARI

a(n)=([0,1; -19,10]^(n-1)*[1;10])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

Sage

[lucas_number1(n,10,19) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009

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)

Extensions

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

Edited by Klaus Brockhaus, Oct 04 2009