A163605 a(n) = ((3+2*sqrt(2))*(5+sqrt(2))^n + (3-2*sqrt(2))*(5-sqrt(2))^n)/2.

3, 19, 121, 773, 4947, 31691, 203129, 1302397, 8352003, 53564899, 343552921, 2203536533, 14133648147, 90655141211, 581477504729, 3729706799437, 23923085385603, 153447597468979, 984245010820921, 6313155366422693

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A163604.

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. A163604.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+2*r)*(5+r)^n+(3-2*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 07 2009

Mathematica

LinearRecurrence[{10, -23}, {3, 19}, 50] (* G. C. Greubel, Jul 29 2017 *)

PARI

x='x+O('x^50); Vec((3-11*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) = 3, a(1) = 19.

G.f.: (3-11*x)/(1-10*x+23*x^2).

E.g.f.: exp(5*x)*( 3*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 07 2009