A164588 a(n) = ((3 + sqrt(18))*(5 + sqrt(8))^n + (3 - sqrt(18))*(5 - sqrt(8))^n)/6.

1, 9, 73, 577, 4529, 35481, 277817, 2174993, 17027041, 133295529, 1043495593, 8168931937, 63949894289, 500627099961, 3919122796697, 30680567267633, 240180585132481, 1880236207775049, 14719292130498313, 115228905772807297, 902061091509601649

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

• nonn

Comments

Binomial transform of A057084. Second binomial transform of A002315. Third binomial transform of A108051 without initial 0. Fourth binomial transform of A096980. Fifth binomial transform of A094015.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-17).

Crossrefs

Cf. A057084, A002315, A108051, A096980, A094015.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+3*r)*(5+2*r)^n+(3-3*r)*(5-2*r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

Mathematica

LinearRecurrence[{10,-17},{1,9},30] (* Harvey P. Dale, Sep 11 2016 *)

PARI

x='x+O('x^50); Vec((1-x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 12 2017

Formula

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

G.f.: (1-x)/(1-10*x+17*x^2).

E.g.f.: (1/3)*exp(5*x)*(3*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Extensions

Extended by Klaus Brockhaus and R. J. Mathar Aug 24 2009