A163611 a(n) = ((5 + 2*sqrt(2))*(5 + sqrt(2))^n + (5 - 2*sqrt(2))*(5 - sqrt(2))^n)/2.

5, 29, 175, 1083, 6805, 43141, 274895, 1756707, 11244485, 72040589, 461782735, 2960893803, 18987935125, 121778793781, 781065429935, 5009742042387, 32132915535365, 206105088378749, 1321993826474095, 8479521232029723

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A163610. Fifth binomial transform of A163888.

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. A163610, A163888.

Programs

Magma

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

Mathematica

LinearRecurrence[{10, -23}, {5, 29}, 50] (* G. C. Greubel, Jul 29 2017 *)

PARI

x='x+O('x^50); Vec((5-21*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Jul 29 2017

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 29.

G.f.: (5-21*x)/(1-10*x+23*x^2).

E.g.f.: exp(5*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Extensions

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