A073357 Binomial transform of tribonacci numbers.

0, 1, 3, 8, 22, 62, 176, 500, 1420, 4032, 11448, 32504, 92288, 262032, 743984, 2112384, 5997664, 17029088, 48350464, 137280832, 389779648, 1106696192, 3142227840, 8921685888

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

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...

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

References

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• Index entries for linear recurrences with constant coefficients, signature (4, -4, 2).

Crossrefs

Cf. A000073, A073313. Trisection of A103685.

Programs

Mathematica

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
LinearRecurrence[{4,-4,2},{0,1,3},30] (* Harvey P. Dale, Nov 13 2011 *)

Formula

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

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

a(n) = A115390(n+1) - A115390(n). - R. J. Mathar, Apr 16 2009