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 A099365 #39 Jan 05 2025 19:51:37
%S A099365 0,1,25,676,18225,491401,13249600,357247801,9632441025,259718659876,
%T A099365 7002771375625,188815108482001,5091005157638400,137268324147754801,
%U A099365 3701153746831741225,99793882840309258276,2690733682941518232225
%N A099365 Squares of A052918(n-1) (generalized Fibonacci).
%C A099365 See the comment in A099279. This is example a=5.
%C A099365 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 5 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 5 kinds of (1/4,1/4)-fences available. - _Michael A. Allen_, Mar 30 2023
%H A099365 G. C. Greubel, <a href="/A099365/b099365.txt">Table of n, a(n) for n = 0..650</a>
%H A099365 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 A099365 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 A099365 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (26,26,-1).
%H A099365 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A099365 a(n) = A052918(n-1)^2, n >= 1, a(0) = 0.
%F A099365 a(n) = 26*a(n-1) + 26*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=25.
%F A099365 a(n) = 27*a(n-1) - a(n-2) - 2*(-1)^n, n >= 2; a(0)=0, a(1)=1.
%F A099365 a(n) = 2*(T(n, 27/2) - (-1)^n)/29 with twice the Chebyshev's T(n, x) polynomials of the first kind. 2*T(n, 27/2) = A090248(n).
%F A099365 G.f.: x*(1-x)/((1-27*x+x^2)*(1+x)) = x*(1-x)/(1-26*x-26*x^2+x^3).
%F A099365 a(n) = (1 - (-1)^n)/2 + 25*Sum_{r=1..n-1} r*a(n-r). - _Michael A. Allen_, Mar 30 2023
%F A099365 Product_{n>=2} (1 + (-1)^n/a(n)) = (5 + sqrt(29))/10 (Falcon, 2016, p. 189, eq. (3.1)). - _Amiram Eldar_, Dec 03 2024
%p A099365 with (combinat):seq(fibonacci(n,5)^2,n=0..16); # _Zerinvary Lajos_, Apr 09 2008
%t A099365 LinearRecurrence[{26,26,-1},{0,1,25},30] (* _Harvey P. Dale_, Sep 25 2019 *)
%o A099365 (Magma) [(2/29)*(Evaluate(ChebyshevFirst(n), 27/2) -(-1)^n): n in [0..30]]; // _G. C. Greubel_, Aug 21 2022
%o A099365 (SageMath)
%o A099365 def A099365(n): return (2/29)*(chebyshev_T(n, 27/2) - (-1)^n)
%o A099365 [A099365(n) for n in (0..30)] # _G. C. Greubel_, Aug 21 2022
%Y A099365 Cf. other squares of k-metallonacci numbers (for k=1 to 10): A007598, A079291, A092936, A099279, this sequence, A099366, A099367, A099369, A099372, A099374.
%K A099365 nonn,easy
%O A099365 0,3
%A A099365 _Wolfdieter Lang_, Oct 18 2004