A135547 Number of cycles for vectors of length n under the Ducci map.

1, 1, 2, 1, 2, 4, 10, 1, 6, 10, 4, 24, 6, 298, 1096, 1, 260, 526, 28, 1098, 16660, 1540, 2050, 2744, 658, 10246, 4870, 599338, 566, 8948416, 34636834, 1, 4198408, 8421892, 4195604, 17043806, 21256, 3538972, 67158052, 35791946, 26214476, 8726292328, 805355524, 806094340, 4296053112, 4297066498, 8388610, 89479864

Offset: 1

Author

N. J. A. Sloane, Feb 24 2008

Keywords

• nonn

Comments

The number of different lengths of cycles is given by A111944 and the maximum length by A038553.

Also, the number of connected components in Arnold's graph G_n associated with the Ducci map. G_n has 2^n vertices, one for each binary vector x. Each node x has a single directed edge which goes from x to y, where y_1 = x_2-x_1, y_2 = x_3-x_2, ..., y_{n-1} = x_n-x_{n-1}, y_n = x_1-x_n. (Since the vectors are binary, we could use here sums instead of differences.)

Remarkably, a(n) = A083843(n) for n=4, 7, 8, 14, 16, 23, 28, 31, 32. - Max Alekseyev, Oct 11 2013

Links

• Max Alekseyev, Table of n, a(n) for n = 1..420
• V. I. Arnold, Complexity of finite sequences of zeros and ones and geometry of finite spaces of functions, Funct. Anal. Other. Math 1(1), 2006, pp. 1-15.
• N. J. Calkin, J. G. Stevens, D. M. Thomas, A characterization for the lengths of cycles of the n-number Ducci game, Fib. Q. 43(1), 2005, 53-59.

Crossrefs

Cf. A038553, A111944

Extensions

a(13)-a(19) from R. J. Mathar, Apr 17 2008

Terms a(20) onward added by Max Alekseyev, Oct 12 2013