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 A099369 #33 Jan 05 2025 19:51:37
%S A099369 0,1,64,4225,278784,18395521,1213825600,80094094081,5284996383744,
%T A099369 348729667233025,23010873040995904,1518368891038496641,
%U A099369 100189335935499782400,6610977802851947141761,436224345652293011573824
%N A099369 Squares of A041025(n-1), n>=1, (generalized Fibonacci).
%C A099369 See the comment in A099279. This is example a=8.
%C A099369 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 8 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 8 kinds of (1/4,1/4)-fences available. - _Michael A. Allen_, Apr 30 2023
%H A099369 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 A099369 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 A099369 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (65,65,-1).
%H A099369 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A099369 a(n) = A041025(n-1)^2, n >= 1, a(0)=0.
%F A099369 a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=64.
%F A099369 a(n) = 66*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2; a(0)=0, a(1)=1.
%F A099369 a(n) = (T(n, 33) - (-1)^n)/34 with the Chebyshev polynomials of the first kind: T(n, 33) = A099370(n).
%F A099369 G.f.: x*(1-x)/((1-66*x+x^2)*(1+x)) = x*(1-x)/(1-65*x-65*x^2+x^3).
%F A099369 a(n) = (1 - (-1)^n)/2 + 64*Sum_{r=1..n-1} r*a(n-r). - _Michael A. Allen_, Apr 30 2023
%F A099369 Product_{n>=2} (1 + (-1)^n/a(n)) = (4 + sqrt(17))/8 (Falcon, 2016, p. 189, eq. (3.1)). - _Amiram Eldar_, Dec 03 2024
%t A099369 LinearRecurrence[{65,65,-1},{0,1,64},20] (* _Harvey P. Dale_, Oct 05 2021 *)
%Y A099369 Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, A099365, A099366, A099367, this sequence, A099372, A099374.
%K A099369 nonn,easy
%O A099369 0,3
%A A099369 _Wolfdieter Lang_, Oct 18 2004