A152939 Number of sets (in the Hausdorff metric geometry) at each location between two sets defining a polygonal configuration consisting of four 4-gonal polygonal components chained with string components of length l as l varies.
29153, 109649, 486385, 2024613, 8634049, 36481021, 154687133, 655020765, 2775107981, 11754906113, 49795616797, 210935942361, 893541701545, 3785099002297, 16033943772281, 67920864283629, 287717416776137, 1218790505711045, 5162879481166789, 21870308363154597
Offset: 1
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.
Programs
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Maple
with(combinat): a := proc(n) local aa, b, c, d, lambda, delta, R, S, F, L, k, m: k:=4: m:=2: 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;
Formula
Conjectures from Colin Barker, Jul 09 2020: (Start)
G.f.: x*(29153 + 22190*x - 17480*x^2 - 4977*x^3) / ((1 + x - x^2)*(1 - 4*x - x^2)).
a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4) for n>4.
(End)