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%I #125 Feb 20 2023 15:17:48
%S 0,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.
%C Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).
%C Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.
%C Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).
%C The partial sum of this sequence is A184985.
%C a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.
%C a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.
%C Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - _Nathaniel Gregg_, Nov 02 2021
%D V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
%D L. H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
%D V. Manturov, Knot Theory, CRC Press, 2004.
%H I. Altintas, <a href="https://doi.org/10.1016/j.amc.2007.04.018">An oriented state model for the Jones polynomial and its applications to alternating links</a>, Appl. Math. Comput. 194 (2007) 168-178.
%H J. A. Baldwin and A. S. Levine, <a href="https://doi.org/10.1016/j.aim.2012.06.006">A combinatorial spanning tree model for knot Floer homology</a>, Advances in Mathematics, Vol. 231 (2012), 1886-1939.
%H A. Banerjee, <a href="http://agnijomaths.com/categories/geometry/topology/knot_theory.html">Knot theory</a> [Foil knot family].
%H D. Denton and P. Doyle, <a href="https://arxiv.org/abs/1106.3545">Shadow movies not arising from knots</a>, arXiv preprint, arXiv:1106.3545 [math.GT], 2011.
%H L. H. Kauffman, <a href="https://doi.org/10.1016/0040-9383(87)90009-7">State models and the Jones polynomial</a>, Topology, Vol. 26 (1987), 395-407.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1902.08989">A generating polynomial for the two-bridge knot with Conway's notation C(n,r)</a>, arXiv:1902.08989 [math.CO], 2019.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F a(n) = ((-1)^2^(n^2 + 3*n + 2) + (-1)^2^(n^2 - n) - (-1)^2^(n^2 - 3*n + 2) + 1)/2.
%F a(n) = (1 + (-1)^((n + 1)!))/2 + Kronecker(n, 2).
%F a(n) = min(n, 3) - 2*(max(n - 2, 0) - max(n - 3, 0)).
%F a(n) = floor(F(n+1)/F(n)) for n > 0, with a(0) = 0, where F(n) = A000045(n) is the n-th Fibonacci number.
%F a(n) = a(n-1) for n > 3, with a(0) = 0, a(1) = 1, a(2) = 2 and a(3) = 1.
%F A005803(a(n)) = A005096(a(n)) = A000007(n).
%F A107583(a(n)) = A103775(n+5).
%F a(n+1) = 2^A185012(n+1), with a(0) = 0.
%F a(n) = A163985(n) mod A004278(n+1).
%F a(n) = A157928(n) + A171386(n+1).
%F a(n) = A063524(n) + A157928(n) + A185012(n).
%F a(n) = A010701(n) - A141044(n) - A179184(n).
%F G.f.: (x + x^2 - x^3)/(1 - x).
%F E.g.f.: (2*exp(x) - 2 + x^2)/2.
%e For n = 2, the shadow of the Hopf link yields 2 two-component state diagrams (see example in A300453). Thus a(2) = 2.
%t CoefficientList[Series[(x + x^2 - x^3)/(1 - x), {x, 0, 100}], x] (* _Wesley Ivan Hurt_, Nov 05 2017 *)
%t f[n_] := If[n > 2, 1, n]; Array[f, 105, 0] (* _Robert G. Wilson v_, Dec 27 2017 *)
%t PadRight[{0,1,2},120,{1}] (* _Harvey P. Dale_, Feb 20 2023 *)
%o (PARI) a(n) = if(n>2, 1, n);
%o (Maxima) makelist((1 + (-1)^((n + 1)!))/2 + kron_delta(n, 2), n, 0, 100);
%Y Cf. A000007, A000045, A004278, A005096, A005803, A010701, A063524, A103775, A107583, A130130, A132925, A141044, A157928, A158411, A163985, A171386, A179184, A184985, A185012, A300453.
%K nonn,easy
%O 0,3
%A _Franck Maminirina Ramaharo_, Nov 05 2017
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