A152928 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 1 as m varies.
113, 765, 5234, 35865, 245813, 1684818, 11547905, 79150509, 542505650, 3718389033, 25486217573, 174685133970, 1197309720209, 8206482907485, 56248070632178, 385530011517753, 2642462009992085, 18111704058426834, 124139466398995745, 850864560734543373
Offset: 2
Links
- Colin Barker, Table of n, a(n) for n = 2..1000
- 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, Q, F, L: F := fibonacci: L := t -> fibonacci(t-1)+fibonacci(t+1): aa := L(2*n)*F(l-2)+F(2*n+2)*F(l-1): b := L(2*n)*F(l-1)+F(2*n+2)*F(l): c := F(2*n+2)*F(l-2)+F(n+2)^2*F(l-1): d := F(2*n+2)*F(l-1)+F(n+2)^2*F(l): Q:=sqrt((d-aa)^2+4*b*c); lambda := (d+aa+Q)/2: delta := (d+aa-Q)/2: : simplify(lambda*((lambda-d)*L(2*n)+b*F(2*n+2))/Q+delta*((lambda-aa)*L(2*n)-b*F(2*n+2))/Q); end proc; # Simplified by M. F. Hasler, Apr 16 2015
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Mathematica
LinearRecurrence[{8, -8, 1}, {113, 765, 5234}, 30] (* Paolo Xausa, Jul 22 2024 *) -
PARI
Vec(x^2*(113 - 139*x + 18*x^2) / ((1 - x)*(1 - 7*x + x^2)) + O(x^20)) \\ Colin Barker, Aug 05 2020
Formula
G.f.: x^2*(113 - 139*x + 18*x^2)/(1 - 8*x + 8*x^2 - x^3). - M. F. Hasler, Apr 16 2015
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>4. - Colin Barker, Aug 05 2020
Extensions
More terms from M. F. Hasler, Apr 16 2015