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A152931 Number of sets (in the Hausdorff metric geometry) at each location between two sets defining a polygonal configuration consisting of three m-gonal polygonal components chained with string components of length 2 as m varies.

%I A152931 #17 Feb 21 2025 22:33:36
%S A152931 4393,80361,1425131,25671393,459934921,8258011407,148150698209,
%T A152931 2658683875329,47706585218947,856070631915129,15361490875216193,
%U A152931 275651271699299271,4946357927482614361,88758815221749418713,1592712152944203460571,28580061055811939151057
%N A152931 Number of sets (in the Hausdorff metric geometry) at each location between two sets defining a polygonal configuration consisting of three m-gonal polygonal components chained with string components of length 2 as m varies.
%H A152931 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 A152931 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (13, 104, -260, -260, 104, 13, -1).
%p A152931 with(combinat): a := proc(n) local aa, b, c, d, lambda, delta, R, S, F, L, k, l: k:=3: l:=2: 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;
%t A152931 LinearRecurrence[{13,104,-260,-260,104,13,-1},{4393,80361,1425131,25671393,459934921,8258011407,148150698209},20] (* _Harvey P. Dale_, Feb 18 2024 *)
%Y A152931 Cf. A152927, A152928, A152929, A152930, A152932, A152933, A152934, A152935.
%K A152931 nonn,easy
%O A152931 2,1
%A A152931 _Steven Schlicker_, Dec 15 2008