A164035 a(n) = ((4+3*sqrt(2))*(5+sqrt(2))^n + (4-3*sqrt(2))*(5-sqrt(2))^n)/4.

2, 13, 84, 541, 3478, 22337, 143376, 920009, 5902442, 37864213, 242885964, 1557982741, 9993450238, 64100899337, 411159637896, 2637275694209, 16916085270482, 108503511738013, 695965156159044, 4464070791616141

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Keywords

• nonn

Comments

Binomial transform of A164034. Fifth binomial transform of A164090.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-23).

Crossrefs

Cf. A164034, A164090.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(5+r)^n+(4-3*r)*(5-r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 09 2009

Mathematica

LinearRecurrence[{10,-23}, {2,13}, 50] (* or *) CoefficientList[Series[(2 - 7*x)/(1 - 10*x + 23*x^2), {x,0,50}], x] (* _G. C. Greubel, Sep 08 2017 *)

PARI

x='x+O('x^50); Vec((2-7*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Sep 08 2017

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 2, a(1) = 13.

G.f.: (2-7*x)/(1-10*x+23*x^2).

E.g.f.: (2*cosh(sqrt(2)*x) + (3*sqrt(2)/2)*sinh(sqrt(2)*x))*exp(5*x). - G. C. Greubel, Sep 08 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 09 2009