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 A007782 Number of factors in the infinite word formed by the Kolakoski sequence A000002. 3

%I

%S 1,2,4,6,10,14,18,26,34,42,50,62,78,94,110,126,142,162,186,218,250,

%T 282,314,346,378,410,446,486,534,590,654,718,782,846,910,974,1038,

%U 1102,1166,1234,1302,1378,1458,1554,1658,1774,1898,2026,2154,2282,2410,2538,2666

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

%C 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)).

%D 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.

%H D. Wilson, <a href="/A007782/b007782.txt">Table of n, a(n) for n = 0..100</a>.

%e 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.

%t 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 *)

%Y Cf. A000002.

%K nonn,nice

%O 0,2

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