A152934 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.
289, 1962, 13429, 92025, 630730, 4323069, 29630737, 203092074, 1392013765, 9541004265, 65395016074, 448224108237, 3072173741569, 21056992082730, 144326770837525, 989230403779929, 6780286055621962, 46472771985573789, 318529117843394545, 2183231052918188010
Offset: 2
Keywords
Links
- S. Schlicker, L. Morales, and D. Schultheis, Polygonal chain sequences in the space of compact sets, J. Integer Seq. 12 (2009), no. 1, Article 09.1.7, 23 pp.
- Index entries for linear recurrences with constant coefficients, signature (8, -8, 1).
Programs
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Maple
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;
Formula
Conjectures from Colin Barker, Jul 09 2020: (Start)
G.f.: x^2*(289 - 350*x + 45*x^2) / ((1 - x)*(1 - 7*x + x^2)).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>4.
(End)