A161947 a(n) = ((4+sqrt(2))*(5+sqrt(2))^n + (4-sqrt(2))*(5-sqrt(2))^n)/4.

2, 11, 64, 387, 2398, 15079, 95636, 609543, 3895802, 24938531, 159781864, 1024232427, 6567341398, 42116068159, 270111829436, 1732448726703, 11111915190002, 71272831185851, 457154262488464, 2932267507610067, 18808127038865998, 120639117713628439, 773804255242366436

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

Keywords

• nonn
• easy

Comments

Fifth binomial transform of A135530.

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..252
• Index entries for linear recurrences with constant coefficients, signature (10,-23).

Crossrefs

Cf. A135530.

Programs

GAP

a := [2, 11];; for n in [3..10^2] do a[n] := 10*a[n-1] - 23*a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+r)*(5+r)^n+(4-r)*(5-r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

Maple

seq(simplify(((4+sqrt(2))*(5+sqrt(2))^n+(4-sqrt(2))*(5-sqrt(2))^n)*1/4), n = 0 .. 20); # Emeric Deutsch, Jun 28 2009

Mathematica

LinearRecurrence[{10,-23}, {2,11}, 50] (* G. C. Greubel, Aug 17 2018 *)
Table[(((4+Sqrt[2])(5+Sqrt[2])^n)+((4-Sqrt[2])(5-Sqrt[2])^n))/4,{n,0,20}]//Simplify (* Harvey P. Dale, Mar 07 2020 *)

PARI

x='x+O('x^30); Vec((2-9*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Aug 17 2018

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n>1; a(0) = 2; a(1) = 11.

G.f.: (2-9*x)/(1-10*x+23*x^2).

E.g.f.: exp(5*x)*(4*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - Stefano Spezia, Oct 24 2023

Extensions

Edited and extended beyond a(4) by Klaus Brockhaus, Jul 01 2009

Extended by Emeric Deutsch, Jun 28 2009