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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.
47
289, 1962, 13429, 92025, 630730, 4323069, 29630737, 203092074, 1392013765, 9541004265, 65395016074, 448224108237, 3072173741569, 21056992082730, 144326770837525, 989230403779929, 6780286055621962, 46472771985573789, 318529117843394545, 2183231052918188010
OFFSET
2,1
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.
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)
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;
KEYWORD
nonn
AUTHOR
Steven Schlicker, Dec 15 2008
STATUS
approved