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

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

Offset: 0

Author

Wolfdieter Lang, Oct 18 2004

Keywords

• nonn
• easy

Comments

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

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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

Formula

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