A038553 Maximum cycle length in differentiation digraph for n-bit binary sequences.

1, 1, 3, 1, 15, 6, 7, 1, 63, 30, 341, 12, 819, 14, 15, 1, 255, 126, 9709, 60, 63, 682, 2047, 24, 25575, 1638, 13797, 28, 475107, 30, 31, 1, 1023, 510, 4095, 252, 3233097, 19418, 4095, 120, 41943, 126, 5461, 1364, 4095, 4094, 8388607, 48, 2097151, 51150, 255, 3276, 3556769739, 27594, 1048575

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Length of longest cycle for vectors of length n under the Ducci map.

Also, the period of polynomial (x+1)^n+1 over GF(2) (cf. A046932). - Max Alekseyev, Oct 12 2013

Per the comment by T. D. Noe originally given in A138006, it appears that for an odd n > 1, a(n) <= n*(2^((n-1)/2)-1). - Max Alekseyev, Jul 10 2025

References

• Simmons, G. J., The structure of the differentiation digraphs of binary sequences. Ars Combin. 35 (1993), A, 71-88. Math. Rev. 95f:05052.

Links

• Max Alekseyev, Table of n, a(n) for n = 1..2458
• Florian Breuer, Igor E. Shparlinski, Lower bounds for periods of Ducci sequences, arXiv:1909.04462 [math.NT], 2019.
• 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 (No. 1, 2005), 53-59.
• O. N. Karpenkov, On examples of difference operators for {0,1}-valued functions over finite sets, Funct. Anal. Other Math. 1 (2006), 175-180. [Gives incorrect value 4095 for a(46).]

Crossrefs

Cf. A111944, A135547, A136043, A119513.

Formula

It appears that whenever b(n) = log2(a(n)/n + 1) is an integer and n > 1, b(n) = A119513(n) = A136043(n). - Andrei Zabolotskii, Jul 28 2025

Extensions

Entry revised by N. J. A. Sloane, Jun 19 2006, Feb 24 2008

a(46) corrected, terms a(51) onward and b-file added by Max Alekseyev, Oct 12 2013

b-file extended by Max Alekseyev, Sep 24 2019