A074297 -id:A074297 - cp's OEIS frontend

A074298 First occurrence of an 'average' valued sequence of length 2n in the Kolakoski sequence (A000002).

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 9, 4, 4, 3, 1, 4, 3, 1, 1, 9, 4, 4, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 3, 1, 1, 4, 3

Offset: 1

Jon Perry, Sep 21 2002

nonn

a(n) is the least k such that K(k) + K(k+1) + ... + K(k + 2*n - 1) = 3*n, where K(m) = A000002(m).

			a[1]=1, as A000002 begins 1,2 (sum 3) a[2]=1, as A000002 begins 1,2,2,1 (sum 6) a[3]=1, as A000002 begins 1,2,2,1,1,2 (sum 9).

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A074296, A074297, A074299.

nmax = 90; a2 = {1, 2, 2}; For[n = 3, n <= 2*nmax, n++, For[i = 1, i <= a2[[n]], i++, AppendTo[a2, 1 + Mod[n - 1, 2]]]]; a[n_] := For[k = 1, True, k++, If[Plus @@ a2[[k ;; k + 2*n - 1]] == 3*n, Return[k]]]; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Sep 25 2012 *)

Offset corrected by Nathaniel Johnston, May 02 2011

A074296 First occurrence of the smallest value subsequence of length n in the Kolakoski sequence (A000002).

Original entry on oeis.org

1, 4, 3, 4, 13, 12, 28, 10, 9, 13, 13, 12, 13, 112, 20, 10, 13, 12, 13, 13, 12, 13, 112, 111, 10, 109, 108, 167, 4, 112, 4, 94, 20, 101, 91, 167, 13, 94, 13, 13, 94, 93, 1511, 91, 90, 157, 743, 94, 750, 776, 775, 217, 743, 742, 743, 743, 742, 173, 217, 216

Offset: 1

Jon Perry, Sep 21 2002

nonn

The sequence of minimal sums begins 1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 15, 17, 18, 19, 21, ...

			a(3) = 3 because the Kolakoski sequence starting at position 3 is 2, 1, 1, which sums to 4, which is the smallest possible sum of 3 consecutive terms.
a(8) = 10 because the Kolakoski sequence starting at position 10 is 1, 2, 2, 1, 1, 2, 1, 1, which sums to 11, which is the smallest possible sum of 8 consecutive values in the Kolakoski sequence. Note that we cannot find a sequence of length eight with a sum of 10 because it would have to be of the form 1, 1, 2, 1, 1, 2, 1, 1, which would mean that 2, 1, 2, 1, 2 would have to appear earlier in the sequence, which would mean that 1, 1, 1 would have to appear even earlier in the sequence, which is impossible.

Cf. A000002, A074297, A074298.

a(8)-a(15) from and edited by Nathaniel Johnston, May 02 2011

More terms from Sean A. Irvine, Jan 18 2025

cp's OEIS Frontend

A074298 First occurrence of an 'average' valued sequence of length 2n in the Kolakoski sequence (A000002).

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