A106803 Expansion of x*(1-x)/(1-2*x-x^2+x^3).

0, 1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209

Offset: 0

Author

Roger L. Bagula, May 17 2005

Keywords

• nonn
• easy
• less

Comments

Essentially a duplicate of A077998: a(n) = A077998(n-1). - Joerg Arndt, Aug 14 2015

a(n) appears in the formula for the nonnegative powers of sigma, the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with rho = 2*cos(Pi/7), the ratio of the smaller heptagon diagonal to the side length, as follows. sigma^n = a(n-1)*1 + B(n)*rho + a(n)*sigma, n>=0, with B(n)=A006054(n). Put a(-1):= 1. See the Steinbach reference, and a comment under A052547.

a(n-1) is the top left entry of the n-th power of the 3X3 matrix [0, 1, 0; 1, 1, 1; 0, 1, 1] or of the 3X3 matrix [0, 0, 1; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2845
• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31.
• Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
• Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Crossrefs

Cf. A006356, A077998, A187068, A187070, A199853.

Programs

Mathematica

m = {{0, 0, 1}, {1, 2, 0}, {1, 1, 0}}; v[0] = {0, 1, 1}; v[n_] := m.v[n - 1]; Table[v[n][[1]], {n, 0, 30}] (* Edited and corrected by L. Edson Jeffery, Oct 18 2017 *)
RecurrenceTable[{a[1]== 0, a[2]== 1, a[3]== 1, a[n]== 2*a[n-1]  + a[n-2] - a[n-3]}, a, {n,30}] (* G. C. Greubel, Aug 14 2015 *)

PARI

concat(0,Vec((1-x)/(x^3-2*x-x^2+1)+O(x^99))) \\ Charles R Greathouse IV, Sep 25 2012

Formula

a(n) = A077998(n-1). - R. J. Mathar, Aug 07 2008

a(n) = A187070(2*n), a(n) = A187068(2*n+2). - L. Edson Jeffery, Mar 10 2011

a(n+1) = - A199853(n+1). - G. C. Greubel, Aug 14 2015

a(n) = 2*a(n-1) + a(n-2) - a(n-3), a(0)=0, a(1)=a(2)=1. - G. C. Greubel, Aug 14 2015

a(n) = A006356(n-2) for n > 1. - Georg Fischer, Oct 21 2018

Extensions

Edited by N. J. A. Sloane, Aug 08 2008