A162769 a(n) = ((1+sqrt(5))*(4+sqrt(5))^n + (1-sqrt(5))*(4-sqrt(5))^n)/2.

1, 9, 61, 389, 2441, 15249, 95141, 593389, 3700561, 23077209, 143911501, 897442709, 5596515161, 34900251489, 217640345141, 1357219994749, 8463716161441, 52780309349289, 329141597018461, 2052549373305509

Offset

0,2

Comments

Binomial transform of A082762. Fourth binomial transform of A162962. Inverse binomial transform of A093145 without initial 0.

Links

Table of n, a(n) for n=0..19.

Index entries for linear recurrences with constant coefficients, signature (8,-11).

Formula

a(n) = 8*a(n-1) - 11*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

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

a(n) = A091870(n)+A091870(n+1). - R. J. Mathar, Feb 04 2021

Mathematica

f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(4 + s)^n + (1 - s)(4 - s)^n)/2]]; Array[f, 21, 0] (* Or *)
a[n_] := 8 a[n - 1] - 11 a[n - 2]; a[0] = 1; a[1] = 9; Array[a, 22, 0] (* Or *)
CoefficientList[Series[(1 + x)/(1 - 8 x + 11 x^2), {x, 0, 21}], x] (* Robert G. Wilson v *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((1+r)*(4+r)^n+(1-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 19 2009

Crossrefs

Cf. A082762, A162962, A093145.

Sequence in context: A001454 A243877 A200674 * A126504 A361280 A025014

Adjacent sequences: A162766 A162767 A162768 * A162770 A162771 A162772

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009

Status

approved