A138513 a(n) = 8*a(n-1) - 5*a(n-2).

1, 3, 19, 137, 1001, 7323, 53579, 392017, 2868241, 20985843, 153545539, 1123435097, 8219753081, 60140849163, 440028027899, 3219519977377, 23556019679521, 172350557549283, 1261024361996659, 9226442108226857

Offset

1,2

Comments

Rightmost digit of each term forms a cycle with period 4: 1, 3, 9, 7, ... (repeat) ...

Limit_{n->oo} a(n)/a(n-1) = 4 + sqrt(11) = 7.31662479...

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (8,-5).

Formula

a(n) = 8*a(n-1) - 5*a(n-2), n> 2; given a(1) = 1, a(2) = 3.

a(n) = upper left term of the 2 X 2 matrix [1,2; 1,7]^n * [1,0].

O.g.f.: x*(1-5*x)/(1-8*x+5*x^2). - R. J. Mathar, Apr 12 2008

a(n) = ((3*sqrt(11)/22 + 1/2)*(4 - sqrt(11))^n + ((-3*sqrt(11)/22 + 1/2)* (4 + sqrt(11))^n. - Emeric Deutsch, Apr 02 2008

Example

a(5) = 1001 = 8*a(4) - 5*a(3) = 8*137 - 5*19.
a(5) = 1001 = upper left term in [1,2; 1,7]^5.

Maple

a[1]:=1: a[2]:=3: for n from 3 to 25 do a[n]:=8*a[n-1]-5*a[n-2] end do: seq(a[n], n=1..20); # Emeric Deutsch, Apr 02 2008

Mathematica

LinearRecurrence[{8, -5}, {1, 3}, 50] (* G. C. Greubel, Sep 28 2017 *)

Prog

(PARI) x='x+O('x^50); Vec(x*(1-5*x)/(1-8*x+5*x^2)) \\ G. C. Greubel, Sep 28 2017

Crossrefs

Sequence in context: A105797 A278189 A221297 * A321515 A094661 A094662

Adjacent sequences: A138510 A138511 A138512 * A138514 A138515 A138516

Keyword

nonn,easy

Author

Gary W. Adamson, Mar 22 2008

Extensions

More terms from R. J. Mathar and Emeric Deutsch, Apr 12 2008

Status

approved