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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.

%I A152928 #21 Jul 22 2024 15:18:34
%S A152928 113,765,5234,35865,245813,1684818,11547905,79150509,542505650,
%T A152928 3718389033,25486217573,174685133970,1197309720209,8206482907485,
%U A152928 56248070632178,385530011517753,2642462009992085,18111704058426834,124139466398995745,850864560734543373
%N 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.
%H A152928 Colin Barker, <a href="/A152928/b152928.txt">Table of n, a(n) for n = 2..1000</a>
%H A152928 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>, J. Integer Seq. 12 (2009), no. 1, Article 09.1.7, 23 pp.
%H A152928 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A152928 G.f.: x^2*(113 - 139*x + 18*x^2)/(1 - 8*x + 8*x^2 - x^3). - _M. F. Hasler_, Apr 16 2015
%F A152928 a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>4. - _Colin Barker_, Aug 05 2020
%p A152928 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
%t A152928 LinearRecurrence[{8, -8, 1}, {113, 765, 5234}, 30] (* _Paolo Xausa_, Jul 22 2024 *)
%o A152928 (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
%Y A152928 Cf. A152927, A152929, A152930, A152931, A152932, A152933, A152934, A152935.
%K A152928 nonn,easy
%O A152928 2,1
%A A152928 _Steven Schlicker_, Dec 15 2008
%E A152928 More terms from _M. F. Hasler_, Apr 16 2015