A260304 a(n) = 5*a(n-1) - 5*a(n-2) for n>1, a(0)=2, a(1)=3.

2, 3, 5, 10, 25, 75, 250, 875, 3125, 11250, 40625, 146875, 531250, 1921875, 6953125, 25156250, 91015625, 329296875, 1191406250, 4310546875, 15595703125, 56425781250, 204150390625, 738623046875, 2672363281250, 9668701171875, 34981689453125, 126564941406250

Offset

0,1

Comments

Lim_{n -> infinity} a(n + 1)/a(n) = 2 + phi = 3.6180339887..., where phi is the golden ratio (A001622).

Links

Table of n, a(n) for n=0..27.

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

Formula

G.f.: (2 - 7*x)/(1 - 5*x + 5*x^2).

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

a(n) = 2*A030191(n) - 7*A030191(n-1). - Bruno Berselli, Nov 23 2015

Mathematica

Table[((5 + 2 Sqrt[5]) ((5 - Sqrt[5])/2)^n + (5 - 2 Sqrt[5]) ((5 + Sqrt[5])/2)^n)/5, {n, 0, 30}]
RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 5 a[n - 1] - 5 a[n - 2]}, a, {n, 0, 30}] (* Bruno Berselli, Nov 23 2015 *)

Prog

(Magma) [n le 2 select n+1 else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 23 2015
(PARI) a(n)=([0, 1; -5, 5]^n*[2; 3])[1, 1] \\ Charles R Greathouse IV, Jul 26 2016

Crossrefs

Cf. A020876, A030191, A098111.

Cf. A093129: initial values 1,2; A081567: initial values 1,3.

Sequence in context: A050837 A107578 A011827 * A259438 A135961 A173253

Adjacent sequences: A260301 A260302 A260303 * A260305 A260306 A260307

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Nov 21 2015

Extensions

Edited by Bruno Berselli, Nov 23 2015

Status

approved