A099373 -id:A099373 - cp's OEIS frontend

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

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

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

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

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).

Cf. A014511.

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

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 *)

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

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)

More terms from Ray Chandler, Nov 06 2003

A099372 a(n) = A099371(n)^2.

Original entry on oeis.org

0, 1, 81, 6724, 558009, 46308025, 3843008064, 318923361289, 26466795978921, 2196425142889156, 182276820063821025, 15126779640154255921, 1255340433312739420416, 104178129185317217638609, 8645529381948016324584129, 717474760572500037722844100, 59541759598135555114671476169

Offset: 0

Wolfdieter Lang, Oct 18 2004

See the comment in A099279. This is example a=9.

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2)-fences if there are 9 kinds of half-square available. A (w,g)-fence is a tile composed of two w X 1 pieces separated horizontally by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,1/4)-fences and (1/4,3/4)-fences if there are 9 kinds of (1/4,1/4)-fence available. - Michael A. Allen, Mar 21 2024

Stefano Spezia, Table of n, a(n) for n = 0..500
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Sergio Falcon, Some series of reciprocal k-Fibonacci numbers, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; ResearchGate link.
Index entries for linear recurrences with constant coefficients, signature (82,82,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A099371, A099373.

Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, A099365, A099366, A099367, A099369, this sequence, A099374.

LinearRecurrence[{82,82,-1},{0,1,81},17] (* Stefano Spezia, Apr 06 2023 *)

a(n) = A099371(n)^2.
a(n) = 82*a(n-1) + 82*a(n-2) - a(n-3), n>=3; a(0)=0, a(1)=1, a(2)=81.
a(n) = 83*a(n-1) - a(n-2) - 2*(-1)^n, n>=2; a(0)=0, a(1)=1.
a(n) = 2*(T(n, 83/2)-(-1)^n)/85 with twice the Chebyshev polynomials of the first kind: 2*T(n, 83/2) = A099373(n).
G.f.: x*(1-x)/((1-83*x+x^2)*(1+x)) = x*(1-x)/(1-82*x-82*x^2+x^3).
E.g.f.: 2*exp(-x)*(exp(85*x/2)*cosh(9*sqrt(85)*x/2) - 1)/85. - Stefano Spezia, Apr 06 2023
a(n) = (1 - (-1)^n)/2 + 81*Sum_{r=1..n-1} r*a(n-r). - Michael A. Allen, Mar 21 2024
Product_{n>=2} (1 + (-1)^n/a(n)) = (9 + sqrt(85))/18 (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

cp's OEIS Frontend

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

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