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%I #41 Sep 08 2022 08:45:24
%S 0,1,3,9,26,74,209,588,1651,4631,12983,36388,101972,285741,800660,
%T 2243445,6286059,17613241,49351342,138279586,387451077,1085614208,
%U 3041824015,8523002359,23880923183,66912861640,187485674652,525323190505,1471922876424,4124236259529
%N Binomial transform of A006053.
%C a(n)/a(n-1) tends to 2.801... = 1 + 2*cos(Pi/7).
%C A(n) := a(n+1)*(-1)^(n+1) appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma, n >= 0, with C(n)= A085810(n)*(-1)^n, and B(n)= A181880(n-2)*(-1)^n. For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.
%C This sequence is constructible as a spiral tiling of similar trapezoids, as follows: start with an isosceles trapezoid with side lengths 3,1,4,1. Each new trapezoid is rotated and scaled so one leg fills all unoccupied space on the short base of the previous trapezoid. a(n) is given by the length of the n-th trapezoid's legs. This process is identical to the recursion relation added by _R. J. Mathar_ in the Formula section. See the Links section for an illustration. - _Andrew B. Hudson_, Jun 19 2019
%H Jinyuan Wang, <a href="/A116423/b116423.txt">Table of n, a(n) for n = 1..1000</a>
%H Andrew B. Hudson, <a href="/A116423/a116423.png">Illustration of the first 7 terms as a spiral tiling of similar trapezoids</a>.
%H Peter Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden Fields: A Case for the Heptagon</a>, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-1)
%F Binomial transform of A006053 starting with A006053(1): (0, 1, 1, 3, 4, 9, 14, ...).
%F From _R. J. Mathar_, Apr 02 2008: (Start)
%F O.g.f.: x^2(1-x)/(1 - 4x + 3x^2 + x^3).
%F a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3). (End)
%e a(5) = 26 = 1*0 + 1*4 + 4*1 + 4*3 + 6*1 = 4 + 4 + 12 + 6 = 26.
%t LinearRecurrence[{4, -3, -1}, {0, 1, 3}, 40] (* _Vincenzo Librandi_, Jul 11 2019 *)
%o (PARI) concat(0, Vec(x^2*(1-x)/(1-4*x+3*x^2+x^3) + O(x^50))) \\ _Michel Marcus_, Sep 13 2014
%o (Magma) I:=[0,1,3]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jul 11 2019
%Y Cf. A006053.
%K nonn
%O 1,3
%A _Gary W. Adamson_, Feb 14 2006
%E More terms from _R. J. Mathar_, Apr 02 2008
%E More terms from _Michel Marcus_, Sep 13 2014
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