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

0,1

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).

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

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 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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
(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
(PARI) x='x+O('x^30); Vec((2-9*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Aug 17 2018

Crossrefs

Cf. A135530.

Sequence in context: A362799 A179120 A038725 * A349023 A001565 A357845

Adjacent sequences: A161944 A161945 A161946 * A161948 A161949 A161950

Keyword

nonn,easy

Author

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

Extensions

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

Extended by Emeric Deutsch, Jun 28 2009

Status

approved