A087798 a(n) = 9*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 9.

2, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, 432083484, 3936182123, 35857722591, 326655685442, 2975758891569, 27108485709563, 246952130277636, 2249677658208287, 20494051054152219, 186696137145578258

Offset: 0

Author

Nikolay V. Kosinov, Dmitry V. Poljakov (kosinov(AT)unitron.com.ua), Oct 10 2003

Keywords

• easy
• nonn

Comments

a(n+1)/a(n) converges to (9 + sqrt(85))/2.

For more information about this type of recurrence follow the Khovanova link and see A054413 and A086902. - Johannes W. Meijer, Jun 12 2010

Examples

			a(4) = 9*a(3) + a(2) = 9*756 + 83 = 6887.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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 (9,1).

Crossrefs

Cf. A014511.

Programs

Magma

I:=[2,9]; [n le 2 select I[n] else 9*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016

Mathematica

RecurrenceTable[{a[0] == 2, a[1] == 9, a[n] == 9 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{9,1}, {2,9}, 30] (* G. C. Greubel, Nov 07 2018 *)

PARI

x='x+O('x^30); Vec((2-9*x)/(1-9*x-x^2)) \\ G. C. Greubel, Nov 07 2018

Formula

a(n) = ((9 + sqrt(85))/2)^n + ((9 - sqrt(85))/2)^n.

G.f.: (2 - 9*x)/(1 - 9*x - x^2). - Philippe Deléham, Nov 02 2008

From Johannes W. Meijer, Jun 12 2010: (Start)

a(2n+1) = 9*A097840(n), a(2n) = A099373(n).

a(3n+1) = A041150(5n), a(3n+2) = A041150(5n+3), a(3n+3) = 2*A041150(5n+4).

Lim_{k->infinity} a(n+k)/a(k) = (A087798(n) + A099371(n)*sqrt(85))/2.

Lim_{n->infinity} A087798(n)/A099371(n) = sqrt(85). (End)

Extensions

More terms from Ray Chandler, Nov 06 2003