This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A099372 #40 Jan 05 2025 19:51:37
%S A099372 0,1,81,6724,558009,46308025,3843008064,318923361289,26466795978921,
%T A099372 2196425142889156,182276820063821025,15126779640154255921,
%U A099372 1255340433312739420416,104178129185317217638609,8645529381948016324584129,717474760572500037722844100,59541759598135555114671476169
%N A099372 a(n) = A099371(n)^2.
%C A099372 See the comment in A099279. This is example a=9.
%C A099372 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
%H A099372 Stefano Spezia, <a href="/A099372/b099372.txt">Table of n, a(n) for n = 0..500</a>
%H A099372 Michael A. Allen and Kenneth Edwards, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%H A099372 Sergio Falcon, <a href="https://ikprress.org/index.php/AJOMCOR/article/view/442">Some series of reciprocal k-Fibonacci numbers</a>, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; <a href="https://www.researchgate.net/publication/297715665_SOME_SERIES_OF_RECIPROCAL_k-FIBONACCI_NUMBERS">ResearchGate link</a>.
%H A099372 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (82,82,-1).
%H A099372 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A099372 a(n) = A099371(n)^2.
%F A099372 a(n) = 82*a(n-1) + 82*a(n-2) - a(n-3), n>=3; a(0)=0, a(1)=1, a(2)=81.
%F A099372 a(n) = 83*a(n-1) - a(n-2) - 2*(-1)^n, n>=2; a(0)=0, a(1)=1.
%F A099372 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).
%F A099372 G.f.: x*(1-x)/((1-83*x+x^2)*(1+x)) = x*(1-x)/(1-82*x-82*x^2+x^3).
%F A099372 E.g.f.: 2*exp(-x)*(exp(85*x/2)*cosh(9*sqrt(85)*x/2) - 1)/85. - _Stefano Spezia_, Apr 06 2023
%F A099372 a(n) = (1 - (-1)^n)/2 + 81*Sum_{r=1..n-1} r*a(n-r). - _Michael A. Allen_, Mar 21 2024
%F A099372 Product_{n>=2} (1 + (-1)^n/a(n)) = (9 + sqrt(85))/18 (Falcon, 2016, p. 189, eq. (3.1)). - _Amiram Eldar_, Dec 03 2024
%t A099372 LinearRecurrence[{82,82,-1},{0,1,81},17] (* _Stefano Spezia_, Apr 06 2023 *)
%Y A099372 Cf. A099371, A099373.
%Y A099372 Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, A099365, A099366, A099367, A099369, this sequence, A099374.
%K A099372 nonn,easy
%O A099372 0,3
%A A099372 _Wolfdieter Lang_, Oct 18 2004