OFFSET
0,3
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).
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).
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Jul 29 2002
STATUS
approved