A007782 Number of factors in the infinite word formed by the Kolakoski sequence A000002.

1, 2, 4, 6, 10, 14, 18, 26, 34, 42, 50, 62, 78, 94, 110, 126, 142, 162, 186, 218, 250, 282, 314, 346, 378, 410, 446, 486, 534, 590, 654, 718, 782, 846, 910, 974, 1038, 1102, 1166, 1234, 1302, 1378, 1458, 1554, 1658, 1774, 1898, 2026, 2154, 2282, 2410, 2538, 2666

Offset

0,2

Comments

a(n) = number of different substrings of length n found in Kolakoski sequence A000002. It is conjectured that a(n) grows like n^(log(3)/log(3/2)).

References

M. Dekking: "What is the long range order in the Kolakoski sequence?" in: The Mathematics of Long-Range Aperiodic Order, ed. R. V. Moody, Kluwer, Dordrecht (1997), pp. 115-125.

Links

D. Wilson, Table of n, a(n) for n = 0..100.

Example

For length 3 only the strings 112, 121, 211, 221, 212, 122 occur, so a(3) = 6. For length 4 only the 10 strings 1121, 1122, 1211, 1212, 1221, 2112, 2121, 2122, 2211, 2212 occur.

Mathematica

nMax = 52; A007782[m_] := A007782[m] = (kolak = {1, 2, 2}; For[n = 3, n <= m, n++, For[k = 1, k <= kolak[[n]], k++, AppendTo[ kolak, 1 + Mod[n - 1, 2]]]]; factors[n_] := Table[ kolak[[k ;; k + n - 1]], {k, 1, Length[kolak] - n + 1}]; Table[ factors[n] // Union // Length, {n, 0, nMax}]); A007782[nMax]; A007782[m = 2*nMax]; While[ A007782[m] != A007782[m/2], m = 2*m]; A007782[m] (* Jean-François Alcover, Jul 24 2013 *)

Crossrefs

Cf. A000002.

Sequence in context: A128422 A309882 A098380 * A035501 A024204 A036641

Adjacent sequences: A007779 A007780 A007781 * A007783 A007784 A007785

Keyword

nonn,nice

Author

Patricia Lamas (lamas(AT)math.uqam.ca)

Extensions

Additional comments from Michael Baake (mbaake(AT)pion09.tphys.physik.uni-tuebingen.de), Feb 19, 2001.

Status

approved