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%I #44 Sep 24 2018 16:53:13
%S 1,3,12,41,146,511,1798,6314,22187,77946,273856,962142,3380337,
%T 11876254,41725295,146595013,515037713,1809501081,6357387289,
%U 22335644540,78472648463,275700866485,968630080476,3403123989780
%N G.f.: 1/(1 - 3 x - 3 x^2 + 4 x^3 + x^4 - x^5).
%C Middle line of 5-wave sequence A038201.
%C Let M denotes the 5 X 5 matrix = row by row (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) and A(n) the vector (x(n),y(n),z(n),t(n),u(n))=M^n*A where A is the vector (1,1,1,1,1) then a(n)=z(n). - _Benoit Cloitre_, Apr 02 2002
%C a(n) appears in the formula for 1/rho(11)^n, with rho(11) := 2*cos(Pi/11) (length ratio (smallest diagonal/side) in the regular 11-gom) when written in the power basis of the degree 5 number field Q(rho(11)): 1/rho(11)^n = a(n)*1 + A230080(n)*rho(11) - A230081(n)*rho(11)^2 - A069006(n-1)* rho(11)^3 + a(n-1)*rho(11)^4, n >= 0, with A069006(-1) = 0 = a(-1). See A230080 with the example for n=4. - _Wolfdieter Lang_, Nov 04 2013
%C From _Wolfdieter Lang_, Nov 20 2013: (Start)
%C The limit a(n+1)/a(n) for n -> infinity is omega(11) := S(4, x) = 1 - 3*x^2 + x^4 with x = rho(11). omega(11) = 1/(2*cos(Pi*5/11)), approx. 3.51333709. For the Chebyshev S-polynomial see A049310. For rho(11) see the preceding comment. The decimal expansion of omega(11) is given in A231186. omega(11) is an integer in Q(rho(11)) with power basis coefficients [1,0,-3,0,1]. It is known to be the length ratio (longest diagonal)/side in the regular 11-gon.
%C This limit follows from the a(n)-recurrence and the solutions of X^5 - 3*X^4 - 3*X^3 + 4*X^2 + X - 1 = 0, which are given by the inverse of the known solutions of the minimal polynomial C(11, x) of rho(11) (see A187360). The other four X solutions are 1/rho(11), with coefficients [3,3,-4,-1,1] in the power basis of Q(rho(11)), approx. 0.52110856, 1/(2*cos(Pi*3/11)) with coefficients [-1,-1,1,0,0], approx. 0.763521119, 1/(2*cos(Pi*7/11)) with coefficients [0,-3,3,1,-1], approx. -1.20361562, and 1/(2*cos(Pi*9/11)) with coefficients [0,1,3,0,-1], approx. -0.59435114. These solutions for X are therefore irrelevant for this sequence.
%C The same limit omega(11) is therefore obtained for the sequences A069006, A230080 and A230081. See the Nov 04 2013 comment.
%C (End)
%D Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
%H F. v. Lamoen, <a href="http://home.wxs.nl/~lamoen/wiskunde/wave.htm">Wave sequences</a>
%H P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), p. 22-31 (formula 5).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-4,-1,1).
%F a(n) = 3a(n-1)+3a(n-2)-4a(n-3)-a(n-4)+a(n-5). Also a(n) = b(4n+2) with b(n) as in 5-wave sequence A038201.
%F G.f.: 1/(1 - 3 x - 3 x^2 + 4 x^3 + x^4 - x^5) = -1/C(11, x), with C(11, x) the minimal polynomial of 2*cos(Pi/11) (see the name and A187360 for C). - _Wolfdieter Lang_, Nov 07 2013
%t b = {-1, 3, 3, -4, -1, 1}; p[x_] := Sum[x^(n - 1)*b[[7 - n]], {n, 1, 6}] q[x_] := ExpandAll[x^5*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 30}], n], {n, 0, 30}] (* _Roger L. Bagula_ and _Gary W. Adamson_, Sep 19 2006 *)
%t LinearRecurrence[{3,3,-4,-1,1},{1,3,12,41,146},30] (* _Harvey P. Dale_, Aug 27 2012 *)
%Y Cf. A006358, A069006, A230080, A230081: same recurrence formula.
%Y Cf. A066170.
%K nonn
%O 0,2
%A _Floor van Lamoen_
%E More terms from _Benoit Cloitre_, Apr 02 2002
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 08 2007
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