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 A099366 #38 Jan 05 2025 19:51:37
%S A099366 0,1,36,1369,51984,1974025,74960964,2846542609,108093658176,
%T A099366 4104712468081,155870980128900,5918992532430121,224765845252215696,
%U A099366 8535183127051766329,324112192982714904804,12307728150216114616225
%N A099366 Squares of A005668.
%C A099366 See the comment in A099279. This is example a=6.
%C A099366 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 6 kinds of half-squares 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 6 kinds of (1/4,1/4)-fences available. - _Michael A. Allen_, Apr 21 2023
%H A099366 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 A099366 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 A099366 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (37,37,-1).
%H A099366 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A099366 a(n) = A005668(n)^2.
%F A099366 a(n) = 37*a(n-1) + 37*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=36.
%F A099366 a(n) = 38*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2; a(0)=0, a(1)=1.
%F A099366 a(n) = (T(n, 19) - (-1)^n)/20 with the Chebyshev polynomials of the first kind: T(n, 19) = A078986(n).
%F A099366 G.f.: x*(1-x)/((1 - 38*x + x^2)*(1+x)) = x*(1-x)/(1 - 37*x - 37*x^2 + x^3).
%F A099366 a(n) = (1 - (-1)^n)/2 + 36*Sum_{r=1..n-1} r*a(n-r). - _Michael A. Allen_, Apr 21 2023
%F A099366 Product_{n>=2} (1 + (-1)^n/a(n)) = (3 + sqrt(10))/6 (Falcon, 2016, p. 189, eq. (3.1)). - _Amiram Eldar_, Dec 03 2024
%p A099366 with (combinat):seq(fibonacci(n,6)^2,n=0..15); # _Zerinvary Lajos_, Apr 09 2008
%t A099366 LinearRecurrence[{37,37,-1},{0,1,36},20] (* _Harvey P. Dale_, Sep 23 2018 *)
%Y A099366 Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, A099365, this sequence, A099367, A099369, A099372, A099374.
%K A099366 nonn,easy
%O A099366 0,3
%A A099366 _Wolfdieter Lang_, Oct 18 2004