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

1,3

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

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)

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)

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A006053.

Sequence in context: A258911 A268093 A127911 * A077845 A291000 A276068

Adjacent sequences: A116420 A116421 A116422 * A116424 A116425 A116426

Keyword

nonn

Author

Gary W. Adamson, Feb 14 2006

Extensions

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

More terms from Michel Marcus, Sep 13 2014

Status

approved