A073357 - OEIS

%I #23 May 07 2017 22:44:04

%S 0,1,3,8,22,62,176,500,1420,4032,11448,32504,92288,262032,743984,

%T 2112384,5997664,17029088,48350464,137280832,389779648,1106696192,

%U 3142227840,8921685888

%N Binomial transform of tribonacci numbers.

%C For n-> infinity the ratio a(n)/a(n-1) approaches 1+c, where c is the real root of the cubic x^3-x^2-x-1=0; c=1.8392867...

%C a(n) = rightmost term of M^n *[100] where M = the 3X3 matrix [1 1 0 / 0 1 1 / 1 1 2]. Middle term of the vector = partial sums of A073357 through a(n-1). E.g., M^5*[1 0 0] = [18 34 62] where 62 = a(5) and 34 = partial sums of A073357 through a(4): 34 = 0+1+3+8+22. - _Gary W. Adamson_, Jul 24 2005

%D Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

%H Harvey P. Dale, <a href="/A073357/b073357.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4, -4, 2).

%F a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3), a(0)=0, a(1)=1, a(2)=3.

%F Generating function A(x)=(x-x^2)/(1-4x+4x^2-2x^3).

%F a(n) = A115390(n+1) - A115390(n). - _R. J. Mathar_, Apr 16 2009

%t h[n_] := h[n]=4*h[n-1]-4*h[n-2]+2*h[n-3]; h[0]=0; h[1]=1; h[2]=3

%t LinearRecurrence[{4,-4,2},{0,1,3},30] (* _Harvey P. Dale_, Nov 13 2011 *)

%Y Cf. A000073, A073313. Trisection of A103685.

%K easy,nonn

%O 0,3

%A Mario Catalani (mario.catalani(AT)unito.it), Jul 29 2002