The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A120747 Sequence relating to the hendecagon (11-gon). 4
 0, 1, 4, 14, 50, 175, 616, 2163, 7601, 26703, 93819, 329615, 1158052, 4068623, 14294449, 50221212, 176444054, 619907431, 2177943781, 7651850657, 26883530748, 94450905714, 331837870408, 1165858298498, 4096053203771, 14390815650209, 50559786403254 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The hendecagon is an 11-sided polygon, see Weisstein. The lengths of the diagonals of the regular hendecagon are r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5, where r[1] = 1 is the length of the edge. The value of limit(a(n)/a(n-1),n=infinity) equals the longest diagonal r[5]. 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]. 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. LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 Jay Kappraff, Slavik Jablan, Gary W. Adamson and Radmila Sazdanovich, Golden Fields, Generalized Fibonacci Sequences and Chaotic Matrices, Forma, Vol. 19 No. 4, pp. 367-387, 2004. P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165 Eric Weisstein's World of Mathematics, Hendecagon. Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1). FORMULA a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5). 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 From Johannes W. Meijer, Aug 03 2011: (Start) 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] Sum_{k=1..5} T(n,k)*r[k] = r[5]^n, n>=1. [Steinbach] r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5. [Kappraff] Sum_{k=1..5} T(n,k) = A006358(n-1). Limit_{n -> 00} T(n,k)/T(n-1,k) = r[5], 1 <= k <= 5. sequence(sequence( T(n,k), k=2..5), n>=1) = A038201(n-4). 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) EXAMPLE From Johannes W. Meijer, Aug 03 2011: (Start) The lengths of the regular hendecagon edge and diagonals are: r[1] = 1.000000000, r[2] = 1.918985948, r[3] = 2.682507066, r[4] = 3.228707416, r[5] = 3.513337092. The first few rows of the T(n,k) array are, n>=1, 1 <= k <=5: 0, 0, 0, 0, 1, ... 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, ... 5, 9, 12, 14, 15, ... 15, 29, 41, 50, 55, ... 55, 105, 146, 175, 190, ... 190, 365, 511, 616, 671, ... (End) MAPLE 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 MATHEMATICA LinearRecurrence[{3, 3, -4, -1, 1}, {0, 1, 4, 14, 50}, 41] (* G. C. Greubel, Nov 13 2022 *) PROG (Magma) R:=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 (SageMath) def A120747_list(prec): P. = PowerSeriesRing(ZZ, prec) return P( x*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) ).list() A120747_list(40) # G. C. Greubel, Nov 13 2022 CROSSREFS Cf. A006358, A033304, A038201, A052963, A065941, A066170. From Johannes W. Meijer, Aug 03 2011: (Start) 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). 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) Sequence in context: A026630 A352456 A034459 * A229314 A055099 A335921 Adjacent sequences: A120744 A120745 A120746 * A120748 A120749 A120750 KEYWORD nonn,easy AUTHOR Gary W. Adamson, Jul 01 2006 EXTENSIONS Edited and information added by Johannes W. Meijer, Aug 03 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 05:37 EST 2023. Contains 367575 sequences. (Running on oeis4.)