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A007782 Number of factors in the infinite word formed by the Kolakoski sequence A000002. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jul 24 2013 *)

CROSSREFS

Cf. A000002.

Sequence in context: A024518 A128422 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

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Last modified February 15 22:28 EST 2019. Contains 320138 sequences. (Running on oeis4.)