OFFSET
0,4
COMMENTS
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).
FORMULA
a(n) = A077998(n-1). - R. J. Mathar, Aug 07 2008
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
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 *)
PROG
(PARI) concat(0, Vec((1-x)/(x^3-2*x-x^2+1)+O(x^99))) \\ Charles R Greathouse IV, Sep 25 2012
CROSSREFS
KEYWORD
nonn,easy,less
AUTHOR
Roger L. Bagula, May 17 2005
EXTENSIONS
Edited by N. J. A. Sloane, Aug 08 2008
STATUS
approved