A090247 a(n) = 26*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 26.

2, 26, 674, 17498, 454274, 11793626, 306180002, 7948886426, 206364867074, 5357537657498, 139089614227874, 3610972432267226, 93746193624720002, 2433790061810452826, 63184795413447053474, 1640370890687812937498

Offset: 0

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004

Keywords

• easy
• nonn

Comments

a(n+1)/a(n) converges to (13+sqrt(168)) =25.9614813... Lim a(n)/a(n+1) as n approaches infinity = 0.0385186... = 1/(13+sqrt(168)) = (13-sqrt(168)). Lim a(n+1)/a(n) as n approaches infinity = 25.9614813... = (13+sqrt(168)) = 1/(13-sqrt(168)). Lim a(n)/a(n+1) = 26 - Lim a(n+1)/a(n).

Examples

			a(4) = 454274 = 26*a(3) - a(2) = 26*17498 - 674 = (13+sqrt(168))^4 + (13-sqrt(168))^4 = 454273.9999977986 + 0.0000022013 = 454274.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..705
• Tanya Khovanova, Recursive Sequences
• Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
• Index entries for linear recurrences with constant coefficients, signature (26,-1).

Crossrefs

Cf. A032000, A019586, A097308.

Programs

Mathematica

a[0] = 2; a[1] = 26; a[n_] := 26a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

Sage

[lucas_number2(n,26,1) for n in range(0,16)] # Zerinvary Lajos, Jun 27 2008

Formula

a(n) = 26a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 26. a(n) = (13+sqrt(168))^n + (13-sqrt(168))^n. (a(n))^2 =a(2n)+2.

G.f.: (2-26*x)/(1-26*x+x^2). - Philippe Deléham, Nov 02 2008

a(n) = 2*A097308(n). - R. J. Mathar, Sep 27 2014