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 A135547 Number of cycles for vectors of length n under the Ducci map. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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. PROG (C++) #include #include #include using namespace std ; int inComp( const vector< set > & comps, const int n) { for(int i=0; i < comps.size() ; i++) if ( comps[i].find(n) != comps[i].end() ) return i; return -1 ; } int firstd(const unsigned i, const int len, const unsigned allbu1, const unsigned hibit) { unsigned d= i ^ (i>>1) ; if ( (i&1) != (i & hibit) >> (len-1) ) d |= hibit ; else d &= allbu1 ; return d ; } int main(int argc, char*argv[]) { for(int n=1;; n++) { vector< set > comps ; unsigned allbu1 = 0 ; for(int i=0 ; i < n-1 ; i++) allbu1 |= (1 << i) ; const unsigned hibit = 1 <<(n-1) ; for(int i=0; i < 1< trac ; for(int ider=i;; ) { int c ; if ( ( c=inComp(comps, ider) ) != -1) { comps[c].insert(ider) ; for(set::const_iterator j=trac.begin() ; j != trac.end() ; j++) comps[c].insert(*j) ; break ; } else if ( trac.find(ider) != trac.end() ) { comps.push_back(trac) ; break ; } else trac.insert(ider) ; ider= firstd(ider, n, allbu1, hibit) ; } } cout << n << " " << comps.size() <

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Last modified September 18 17:02 EDT 2020. Contains 337170 sequences. (Running on oeis4.)