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A152929 Number of sets (in the Hausdorff metric geometry) at each location between two sets defining a polygonal configuration consisting of two 4-gonal polygonal components chained with string components of length l as l varies.

%I A152929 #37 Jul 23 2024 14:57:46
%S A152929 113,176,289,465,754,1219,1973,3192,5165,8357,13522,21879,35401,57280,
%T A152929 92681,149961,242642,392603,635245,1027848,1663093,2690941,4354034,
%U A152929 7044975,11399009,18443984,29842993,48286977,78129970,126416947,204546917,330963864,535510781,866474645
%N A152929 Number of sets (in the Hausdorff metric geometry) at each location between two sets defining a polygonal configuration consisting of two 4-gonal polygonal components chained with string components of length l as l varies.
%H A152929 Colin Barker, <a href="/A152929/b152929.txt">Table of n, a(n) for n = 1..1000</a>
%H A152929 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 A152929 P. E. Weidmann, <a href="http://worldwidemann.com/the-sequencer-oeis-survey/#a152929httpoeisorga152929">The OEIS Sequencer survey</a>, Apr 11 2015.
%H A152929 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A152929 a(n) = (163*A000045(n)+63*A000032(n))/2. - Conjectured by _Philipp Emanuel Weidmann_, cf. LINKS.
%F A152929 G.f.: x*(113 + 63*x)/(1 - x - x^2). - _M. F. Hasler_, Apr 16 2015
%F A152929 a(n) = a(n-1) + a(n-2) for n>2. - _Colin Barker_, Aug 05 2020
%F A152929 a(n) = Lucas(n+9) - Fibonacci(n+6) - Fibonacci(n-5). - _Greg Dresden_, Mar 14 2022
%p A152929 with(combinat): a := proc(n) local aa, b, c, d, lambda, delta, R, S, F, L4, Q: F := fibonacci: L4 := F(3)+F(5): aa := L4*F(n-2)+F(6)*F(n-1): b := L4*F(n-1)+F(6)*F(n): c := F(6)*F(n-2)+F(4)^2*F(n-1): d := F(6)*F(n-1)+F(4)^2*F(n): Q := sqrt((d-aa)^2+4*b*c); lambda := (d+aa+Q)/2: delta := (d+aa-Q)/2: R := ((lambda-d)*L4+b*F(6))/Q: S := ((lambda-aa)*L4-b*F(6))/Q: simplify(R*lambda+S*delta); end proc: # Simplified by _M. F. Hasler_, Apr 16 2015
%t A152929 LinearRecurrence[{1, 1}, {113, 176}, 50] (* _Paolo Xausa_, Jul 23 2024 *)
%o A152929 (PARI) A152929(n)=50*fibonacci(n)+63*fibonacci(n+1) \\ _M. F. Hasler_, Apr 14 2015
%o A152929 (PARI) Vec(x*(113 + 63*x) / (1 - x - x^2) + O(x^30)) \\ _Colin Barker_, Aug 05 2020
%Y A152929 Cf. A000032, A000045.
%Y A152929 Cf. A152927, A152928, A152930, A152931, A152932, A152933, A152934, A152935.
%K A152929 nonn,easy
%O A152929 1,1
%A A152929 _Steven Schlicker_, Dec 15 2008
%E A152929 More terms from _M. F. Hasler_, Apr 16 2015