A116423 Binomial transform of A006053.

0, 1, 3, 9, 26, 74, 209, 588, 1651, 4631, 12983, 36388, 101972, 285741, 800660, 2243445, 6286059, 17613241, 49351342, 138279586, 387451077, 1085614208, 3041824015, 8523002359, 23880923183, 66912861640, 187485674652, 525323190505, 1471922876424, 4124236259529

Offset: 1

Author

Gary W. Adamson, Feb 14 2006

Keywords

• nonn

Comments

a(n)/a(n-1) tends to 2.801... = 1 + 2*cos(Pi/7).

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.

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

Examples

			a(5) = 26 = 1*0 + 1*4 + 4*1 + 4*3 + 6*1 = 4 + 4 + 12 + 6 = 26.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000
• Andrew B. Hudson, Illustration of the first 7 terms as a spiral tiling of similar trapezoids.
• Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-1)

Crossrefs

Cf. A006053.

Programs

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

Mathematica

LinearRecurrence[{4, -3, -1}, {0, 1, 3}, 40] (* Vincenzo Librandi, Jul 11 2019 *)

PARI

concat(0, Vec(x^2*(1-x)/(1-4*x+3*x^2+x^3) + O(x^50))) \\ Michel Marcus, Sep 13 2014

Formula

Binomial transform of A006053 starting with A006053(1): (0, 1, 1, 3, 4, 9, 14, ...).

From R. J. Mathar, Apr 02 2008: (Start)

O.g.f.: x^2(1-x)/(1 - 4x + 3x^2 + x^3).

a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3). (End)

Extensions

More terms from R. J. Mathar, Apr 02 2008

More terms from Michel Marcus, Sep 13 2014