%I #18 Jul 03 2023 10:51:29
%S 32733,80361,215658,559305,1469565,3842082,10063989,26342577,68971050,
%T 180563265,472726053,1237607586,3240104013,8482697145,22207994730,
%U 58141279737,152215851789,398506268322,1043302960485,2731402605825,7150904864298,18721311979761
%N Number of sets (in the Hausdorff metric geometry) at each location between two sets defining a polygonal configuration consisting of three 6-gonal polygonal components chained with string components of length l as l 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>, JIS 12 (2009) 09.1.7.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2, 2, -1).
%F Conjectures from _Colin Barker_, Jul 09 2020: (Start)
%F G.f.: 9*x*(3637 + 1655*x - 1170*x^2) / ((1 + x)*(1 - 3*x + x^2)).
%F a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) for n>3.
%F (End)
%p with(combinat): a := proc(n) local aa, b, c, d, lambda, delta, R, S, F, L, k, m: k:=3: m:=3: F := t -> fibonacci(t): L := t -> fibonacci(t-1)+fibonacci(t+1): aa := (m, n) -> L(2*m)*F(n-2)+F(2*m+2)*F(n-1): b := (m, n) -> L(2*m)*F(n-1)+F(2*m+2)*F(n): c := (m, n) -> F(2*m+2)*F(n-2)+F(m+2)^2*F(n-1): d := (m, n) -> F(2*m+2)*F(n-1)+F(m+2)^2*F(n): lambda := (m,n) -> (d(m, n)+aa(m, n)+sqrt((d(m, n)-aa(m, n))^2+4*b(m, n)*c(m, n)))*(1/2): delta := (m,n) -> (d(m, n)+aa(m, n)-sqrt((d(m, n)-aa(m, n))^2+4*b(m, n)*c(m, n)))*(1/2): R := (m,n) -> ((lambda(m, n)-d(m, n))*L(2*m)+b(m, n)*F(2*m+2))/(2*lambda(m, n)-d(m, n)-aa(m, n)): S := (m,n) -> ((lambda(m, n)-aa(m, n))*L(2*m)-b(m, n)*F(2*m+2))/(2*lambda(m, n)-d(m, n)-aa(m, n)): simplify(R(m, n)*lambda(m, n)^(k-1)+S(m, n)*delta(m, n)^(k-1)); end proc;
%Y Cf. A152927, A152928, A152929, A152930, A152931, A152933, A152934, A152935.
%K nonn
%O 1,1
%A _Steven Schlicker_, Dec 15 2008