A090300 a(n) = 14*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 14.

2, 14, 198, 2786, 39202, 551614, 7761798, 109216786, 1536796802, 21624372014, 304278004998, 4281516441986, 60245508192802, 847718631141214, 11928306344169798, 167844007449518386, 2361744410637427202

Offset: 0

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Keywords

• easy
• nonn

Comments

a(n+1)/a(n) converges to (7+sqrt(50)) = 14.071067811...

Lim_{n->infinity} a(n)/a(n+1) = 0.071067811... = 1/(7+sqrt(50)) = sqrt(50) - 7.

Lim_{n->infinity} a(n+1)/a(n) = 14.071067811... = (7+sqrt(50)) = 1/(sqrt(50) - 7).

Examples

			a(4) = 39202 = 14*a(3) + a(2) = 14*2786 + 198 = (7+sqrt(50))^4 + (7-sqrt(50))^4 = 39201.999974491 + 0.000025508 = 39202.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..870
• 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 (14, 1).

Crossrefs

Cf. A050012.

Programs

Mathematica

LinearRecurrence[{14,1},{2,14},20] (* Harvey P. Dale, Jul 12 2020 *)

Formula

a(n) = 14*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 14.

a(n) = (7+sqrt(50))^n + (7-sqrt(50))^n.

(a(n))^2 = a(2n)-2 if n = 1, 3, 5, ...; (a(n))^2 = a(2n)+2 if n = 2, 4, 6, ....

G.f.: (2-14*x)/(1-14*x-x^2). - Philippe Deléham, Nov 02 2008

Extensions

More terms from Ray Chandler, Feb 14 2004