A163073 - OEIS
A163073
a(n) = ((5+sqrt(5))*(4+sqrt(5))^n + (5-sqrt(5))*(4-sqrt(5))^n)/10.
4
1, 5, 29, 177, 1097, 6829, 42565, 265401, 1654993, 10320533, 64359341, 401348865, 2502838169, 15607867837, 97331722837, 606967236489, 3785088940705, 23604071924261, 147196597046333, 917927985203793, 5724261314120681, 35696882675723725, 222608186950462309, 1388199786170737497
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
MATHEMATICA
LinearRecurrence[{8, -11}, {1, 5}, 30] (* Harvey P. Dale, Dec 11 2017 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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
(PARI) x='x+O('x^30); Vec((1-3*x)/(1-8*x+11*x^2)) \\ G. C. Greubel, Jan 08 2018
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009