A099868 a(n) = 5*a(n-1) - a(n-2), a(0) = 3, a(1) = 25.

3, 25, 122, 585, 2803, 13430, 64347, 308305, 1477178, 7077585, 33910747, 162476150, 778470003, 3729873865, 17870899322, 85624622745, 410252214403, 1965636449270, 9417930031947, 45124013710465, 216202138520378, 1035886678891425, 4963231255936747

Offset: 0

Author

Creighton Dement, Oct 28 2004

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (5,-1).

Crossrefs

Cf. A003501, A004254.

Programs

GAP

a:=[3,25];; for n in [3..30] do a[n]:=5*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Nov 20 2018

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (3 +10*x)/(1-5*x+x^2))); // G. C. Greubel, Nov 20 2018

Maple

a[0]:=3: a[1]:=25: for n from 2 to 30 do a[n]:=5*a[n-1]-a[n-2] od: seq(a[n],n=0..25);

Mathematica

LinearRecurrence[{5,-1}, {3,25}, 30] (* G. C. Greubel, Nov 20 2018 *)

PARI

Vec((3+10*x) / (1-5*x+x^2) + O(x^30)) \\ Colin Barker, Mar 28 2017

Sage

s=((3+10*x)/(1-5*x+x^2)).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 20 2018

Formula

|2*A099867(n) + a(n) - A003501(n+1)| = 20*A004254(n)

G.f.: (3 + 10*x) / (1 - 5*x + x^2). - Emeric Deutsch, Dec 03 2004

a(n) = (2^(-1-n)*((5-sqrt(21))^n*(-35+3*sqrt(21)) + (5+sqrt(21))^n*(35+3*sqrt(21)))) / sqrt(21). - Colin Barker, Mar 28 2017