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%I #21 Jul 03 2023 10:54:04
%S 289,1962,13429,92025,630730,4323069,29630737,203092074,1392013765,
%T 9541004265,65395016074,448224108237,3072173741569,21056992082730,
%U 144326770837525,989230403779929,6780286055621962,46472771985573789,318529117843394545,2183231052918188010
%N Number of sets (in the Hausdorff metric geometry) at each location between two sets defining a polygonal configuration consisting of two m-gonal polygonal components chained with string components of length 3 as m varies.
%H S. Schlicker, L. Morales, and D. Schultheis, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Schlicker/schlicker.html">Polygonal chain sequences in the space of compact sets</a>, J. Integer Seq. 12 (2009), no. 1, Article 09.1.7, 23 pp.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8, -8, 1).
%F Conjectures from _Colin Barker_, Jul 09 2020: (Start)
%F G.f.: x^2*(289 - 350*x + 45*x^2) / ((1 - x)*(1 - 7*x + x^2)).
%F a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>4.
%F (End)
%p with(combinat): a := proc(n) local aa, b, c, d, lambda, delta, R, S, F, L, k, l: k:=2: l:=3: F := t -> fibonacci(t): L := t -> fibonacci(t-1)+fibonacci(t+1): aa := (n, l) -> L(2*n)*F(l-2)+F(2*n+2)*F(l-1): b := (n, l) -> L(2*n)*F(l-1)+F(2*n+2)*F(l): c := (n, l) -> F(2*n+2)*F(l-2)+F(n+2)^2*F(l-1): d := (n, l) -> F(2*n+2)*F(l-1)+F(n+2)^2*F(l): lambda := (n,l) -> (d(n, l)+aa(n, l)+sqrt((d(n, l)-aa(n, l))^2+4*b(n, l)*c(n, l)))*(1/2): delta := (n,l) -> (d(n, l)+aa(n, l)-sqrt((d(n, l)-aa(n, l))^2+4*b(n, l)*c(n, l)))*(1/2): R := (n,l) -> ((lambda(n, l)-d(n, l))*L(2*n)+b(n, l)*F(2*n+2))/(2*lambda(n, l)-d(n, l)-aa(n, l)): S := (n,l) -> ((lambda(n, l)-aa(n, l))*L(2*n)-b(n, l)*F(2*n+2))/(2*lambda(n, l)-d(n, l)-aa(n, l)): simplify(R(n, l)*lambda(n, l)^(k-1)+S(n, l)*delta(n, l)^(k-1)); end proc;
%Y Cf. A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152935.
%K nonn
%O 2,1
%A _Steven Schlicker_, Dec 15 2008
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