cp's OEIS Frontend

A120747 Sequence relating to the 11-gon (or hendecagon).

%I A120747 #31 Aug 06 2024 22:29:07
%S A120747 0,1,4,14,50,175,616,2163,7601,26703,93819,329615,1158052,4068623,
%T A120747 14294449,50221212,176444054,619907431,2177943781,7651850657,
%U A120747 26883530748,94450905714,331837870408,1165858298498,4096053203771,14390815650209,50559786403254
%N A120747 Sequence relating to the 11-gon (or hendecagon).
%C A120747 The hendecagon is an 11-sided polygon. The preferred word in the OEIS is 11-gon.
%C A120747 The lengths of the diagonals of the regular 11-gon are r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5, where r[1] = 1 is the length of the edge.
%C A120747 The value of limit(a(n)/a(n-1),n=infinity) equals the longest diagonal r[5].
%C A120747 The a(n) equal the matrix elements M^n[1,2], where M = Matrix([[1,1,1,1,1], [1,1,1,1,0], [1,1,1,0,0], [1,1,0,0,0], [1,0,0,0,0]]). The characteristic polynomial of M is (x^5 - 3x^4 - 3x^3 + 4x^2 + x - 1) with roots x1 = -r[4]/r[3], x2 = -r[2]/r[4], x3 = r[1]/r[2], x4 = r[3]/r[5] and x5 = r[5]/r[1].
%C A120747 Note that M^4*[1,0,0,0,0] = [55, 50, 41, 29, 15] which are all terms of the 5-wave sequence A038201. This is also the case for the terms of M^n*[1,0,0,0,0], n>=1.
%H A120747 G. C. Greubel, <a href="/A120747/b120747.txt">Table of n, a(n) for n = 1..1000</a>
%H A120747 Jay Kappraff, Slavik Jablan, Gary W. Adamson and Radmila Sazdanovich, <a href="https://archive.bridgesmathart.org/2005/bridges2005-369.pdf">Golden Fields, Generalized Fibonacci Sequences and Chaotic Matrices</a>, Forma, Vol. 19 No. 4, pp. 367-387, 2004.
%H A120747 P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31, <a href="http://www.ams.org/mathscinet-getitem?mr=1439165">MR 1439165</a>
%H A120747 Eric Weisstein's World of Mathematics, <a href="http://www.mathworld.wolfram.com/Hendecagon.html">Hendecagon</a>.
%H A120747 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-4,-1,1).
%F A120747 a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5).
%F A120747 G.f.: x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
%F A120747 From _Johannes W. Meijer_, Aug 03 2011: (Start)
%F A120747 a(n) = T(n,4) with T(n,k) = Sum_{k1 = 6-k..6} T(n-1, k1), T(1,1) = T(1,2) = T(1,3) = T(1,4) = 0 and T(1,5) = 1, n>=1 and 1 <= k <= 5. [Steinbach]
%F A120747 Sum_{k=1..5} T(n,k)*r[k] = r[5]^n, n>=1. [Steinbach]
%F A120747 r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5. [Kappraff]
%F A120747 Sum_{k=1..5} T(n,k) = A006358(n-1).
%F A120747 Limit_{n -> 00} T(n,k)/T(n-1,k) = r[5], 1 <= k <= 5.
%F A120747 sequence(sequence( T(n,k), k=2..5), n>=1) = A038201(n-4).
%F A120747 G.f.: (x^2*(x - x1)*(x - x2))/((x - x3)*(x - x4)*(x - x5)*(x - x6)*(x - x7)) with x1 = phi, x2 = (1-phi), x3 = r[1] - r[3], x4 = r[3] - r[5], x5 = r[5] - r[4], x6 = r[4] - r[2], x7 = r[2], where phi = (1 + sqrt(5))/2 is the golden ratio A001622. (End)
%e A120747 From _Johannes W. Meijer_, Aug 03 2011: (Start)
%e A120747 The lengths of the regular hendecagon edge and diagonals are:
%e A120747   r[1] = 1.000000000, r[2] = 1.918985948, r[3] = 2.682507066,
%e A120747   r[4] = 3.228707416, r[5] = 3.513337092.
%e A120747 The first few rows of the T(n,k) array are, n>=1, 1 <= k <=5:
%e A120747     0,   0,   0,   0,   1, ...
%e A120747     1,   1,   1,   1,   1, ...
%e A120747     1,   2,   3,   4,   5, ...
%e A120747     5,   9,  12,  14,  15, ...
%e A120747    15,  29,  41,  50,  55, ...
%e A120747    55, 105, 146, 175, 190, ...
%e A120747   190, 365, 511, 616, 671, ... (End)
%p A120747 nmax:=27: m:=5: for k from 1 to m-1 do T(1,k):=0 od: T(1,m):=1: for n from 2 to nmax do for k from 1 to m do T(n,k):= add(T(n-1,k1), k1=m-k+1..m) od: od: for n from 1 to nmax/3 do seq(T(n,k), k=1..m) od; for n from 1 to nmax do a(n):=T(n,4) od: seq(a(n), n=1..nmax); # _Johannes W. Meijer_, Aug 03 2011
%t A120747 LinearRecurrence[{3, 3, -4, -1, 1}, {0, 1, 4, 14, 50}, 41] (* _G. C. Greubel_, Nov 13 2022 *)
%o A120747 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) )); // _G. C. Greubel_, Nov 13 2022
%o A120747 (SageMath)
%o A120747 def A120747_list(prec):
%o A120747     P.<x> = PowerSeriesRing(ZZ, prec)
%o A120747     return P( x*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) ).list()
%o A120747 A120747_list(40) # _G. C. Greubel_, Nov 13 2022
%Y A120747 Cf. A006358, A033304, A038201, A052963, A065941, A066170.
%Y A120747 From _Johannes W. Meijer_, Aug 03 2011: (Start)
%Y A120747 Cf. A006358 (T(n+2,1) and T(n+1,5)), A069006 (T(n+1,2)), A038342 (T(n+1,3)), this sequence (T(n,4)) (m=5: hendecagon or 11-gon).
%Y A120747 Cf. A000045 (m=2; pentagon or 5-gon); A006356, A006054 and A038196 (m=3: heptagon or 7-gon); A006357, A076264, A091024 and A038197 (m=4: enneagon or 9-gon); A006359, A069007, A069008, A069009, A070778 (m=6; tridecagon or 13-gon); A025030 (m=7: pentadecagon or 15-gon); A030112 (m=8: heptadecagon or 17-gon). (End)
%K A120747 nonn,easy
%O A120747 1,3
%A A120747 _Gary W. Adamson_, Jul 01 2006
%E A120747 Edited and information added by _Johannes W. Meijer_, Aug 03 2011