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 A106803 Expansion of x*(1-x)/(1-2*x-x^2+x^3). 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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). 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 MATHEMATICA m = {{0, 0, 1}, {1, 2, 0}, {1, 1, 0}}; v = {0, 1, 1}; v[n_] := m.v[n - 1]; Table[v[n][], {n, 0, 30}] (* Edited and corrected by L. Edson Jeffery, Oct 18 2017 *) RecurrenceTable[{a== 0, a== 1, a== 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 Cf. A006356, A077998, A187068, A187070, A199853. Sequence in context: A339154 A063119 A218982 * A199853 A006356 A077998 Adjacent sequences:  A106800 A106801 A106802 * A106804 A106805 A106806 KEYWORD nonn,easy,less AUTHOR Roger L. Bagula, May 17 2005 EXTENSIONS Edited by N. J. A. Sloane, Aug 08 2008 STATUS approved

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Last modified June 21 04:10 EDT 2021. Contains 345354 sequences. (Running on oeis4.)