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, 1700759285364356541, 15493529705424787127, 141142526634187440684

Offset

0,1

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

From Klaus Purath, Sep 13 2025: (Start)

Let (t) be any recurrence of the form (9,1) including this sequence. Then the following always applies: a(n) = (t(i+2*n) + (-1)^n*t(i))/t(i+n) regardless of initial values provided that t(i+n) != 0 for any integer i and n >= 1.

For any three consecutive terms (x, y, z) applies |y^2 - x*z| = GCD(a(n-1) + a(n+1)) = 85 and (a(n-1) + a(n+1))/85 = A099371(n). (End)

Links

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

Tanya Khovanova, Recursive Sequences.

Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for linear recurrences with constant coefficients, signature (9,1).

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).

Limit_{k->oo} a(n+k)/a(k) = (a(n) + A099371(n)*sqrt(85))/2.

Limit_{n->oo} a(n)/A099371(n) = sqrt(85). (End)

E.g.f.: 2*exp(9*x/2)*cosh(sqrt(85)*x/2). - Stefano Spezia, Dec 21 2025

From Klaus Purath, Jan 14 2026: (Start)

a(4*n+1) = a(2*n)*a(2*n+1) - 9.

a(4*n+3) = a(2*n+1)*a(2*n+2) + 9. (End)

Example

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

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 *)

Prog

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

Crossrefs

Sequence in context: A006040 A391521 A067309 * A113146 A323769 A375838

Adjacent sequences: A087795 A087796 A087797 * A087799 A087800 A087801

Keyword

easy,nonn

Author

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

Extensions

More terms from Ray Chandler, Nov 06 2003

Status

approved