A163616 a(n) = ((1 + 3*sqrt(2))*(5 + sqrt(2))^n + (1 - 3*sqrt(2))*(5 - sqrt(2))^n)/2.

1, 11, 87, 617, 4169, 27499, 179103, 1158553, 7466161, 48014891, 308427207, 1979929577, 12705470009, 81516319819, 522937387983, 3354498523993, 21517425316321, 138020787111371, 885307088838327, 5678592784821737

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A163615. Fifth binomial transform of A163864. Inverse binomial transform of A081183 without initial 0.

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. A163615, A163864, A081183.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(5+r)^n+(1-3*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

Mathematica

CoefficientList[Series[(1 + x)/(1 - 10 x + 23 x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jun 14 2014 *)
LinearRecurrence[{10, -23}, {1, 11}, 50] (* G. C. Greubel, Jul 30 2017 *)

PARI

x='x+O('x^50); Vec((1+x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Jul 30 2017

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 11.

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

E.g.f.: exp(5*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017

a(n) = A081182(n)+A081182(n+1). - R. J. Mathar, Jul 01 2022

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009