A163073 a(n) = ((5+sqrt(5))*(4+sqrt(5))^n + (5-sqrt(5))*(4-sqrt(5))^n)/10.

1, 5, 29, 177, 1097, 6829, 42565, 265401, 1654993, 10320533, 64359341, 401348865, 2502838169, 15607867837, 97331722837, 606967236489, 3785088940705, 23604071924261, 147196597046333, 917927985203793, 5724261314120681, 35696882675723725, 222608186950462309, 1388199786170737497

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A082761. Fourth binomial transform of A074872.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-11).

Crossrefs

Cf. A082761, A074872 (1,1,5,5,25,25,...).

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((5+r)*(4+r)^n+(5-r)*(4-r)^n)/10: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 24 2009

Mathematica

LinearRecurrence[{8,-11},{1,5},30] (* Harvey P. Dale, Dec 11 2017 *)

PARI

x='x+O('x^30); Vec((1-3*x)/(1-8*x+11*x^2)) \\ G. C. Greubel, Jan 08 2018

Formula

a(n) = 8*a(n-1)-11*a(n-2) for n > 1; a(0) = 1, a(1) = 5.

G.f.: (1-3*x)/(1-8*x+11*x^2).

E.g.f.: exp(4*x)*(5*cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, Oct 25 2023

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 24 2009