A187199
Parse the Kolakoski sequence A000002 into distinct phrases 1, 2, 21, 12, 122, 1221, 121, 12212, 11, ...; a(n) = length of n-th phrase.
Original entry on oeis.org
1, 1, 2, 2, 3, 4, 3, 5, 2, 3, 3, 4, 6, 3, 6, 4, 2, 5, 4, 6, 4, 3, 5, 7, 4, 5, 7, 5, 4, 6, 6, 5, 5, 6, 6, 4, 8, 5, 5, 9, 10, 7, 9, 8, 7, 5, 8, 11, 5, 6, 6, 7, 8, 7, 6, 6, 7, 7, 7, 9, 4, 4, 8, 8, 10, 5, 7, 8, 7, 9, 8, 12, 6, 10, 6, 8, 6, 10, 7, 9, 9, 8, 7, 8, 7, 9, 8, 7, 8, 8, 9, 9, 10, 8, 10, 9, 8, 11, 5, 6
Offset: 1
A376677
List of subwords (or factors) of the Kolakoski sequence (A000002).
Original entry on oeis.org
1, 2, 11, 12, 21, 22, 112, 121, 122, 211, 212, 221, 1121, 1122, 1211, 1212, 1221, 2112, 2121, 2122, 2211, 2212, 11211, 11212, 11221, 12112, 12122, 12211, 12212, 21121, 21122, 21211, 21221, 22112, 22121, 22122, 112112, 112122, 112212, 121121, 121122, 121221
Offset: 1
A283511
Length of shortest prefix of the Kolakoski sequence K (A000002) containing all blocks of length n that appear in K.
Original entry on oeis.org
2, 5, 8, 22, 23, 25, 156, 157, 158, 159, 306, 356, 357, 358, 359, 360, 503, 690, 1432, 1433, 1434, 1435, 1436, 1437, 1438, 3014, 4507, 6280, 7726, 7727, 7728, 7729, 7730, 7731, 7732, 7733, 7734, 7735, 16857, 16858, 16859, 30365, 30366, 30367, 30368, 30369, 30370, 30371, 30372, 30373, 30374, 30375
Offset: 1
The Kolakoski sequence contains 10 distinct blocks of length 4; the last to appear is 2121, which appears for the first time beginning at position 19 of K (indexing starting at position 1), so a prefix of length 22 contains all 10 length-4 subwords.
A376638
a(n) is the number of n-digit numbers in A376637.
Original entry on oeis.org
2, 4, 4, 6, 8, 4, 12, 8, 8, 12, 20, 0, 16, 16, 12, 24, 16, 8, 12, 16, 32, 12, 20, 32, 24, 16, 16, 12, 16, 20, 40, 40, 8, 28, 44, 20, 44, 28, 20, 24, 20, 20, 16, 32, 20, 44, 44, 56, 28, 28, 20, 60, 52, 24, 56, 56, 20, 36, 36, 32, 24, 24, 32, 24, 16, 60, 16, 32
Offset: 1
Sequence A376637 begins: 1, 2, 11, 12, 21, 22, 112, 122, 211, 221, 1121.
So a(1) = 2, a(2) = 4, a(3) = 4.
A283512
Last block of length-n to appear for the first time in the Kolakoski sequence K (A000002).
Original entry on oeis.org
2, 11, 212, 2121, 21211, 121121, 2212212, 22122121, 221221211, 2212212112, 22122121122, 112212211211, 1122122112112, 11221221121121, 112212211211212, 1122122112112122, 11211212212211211, 112212211211212211, 1211212212112212212, 12112122121122122121, 121121221211221221211
Offset: 1
For n = 4 the last block of length 4 to appear is 2121.
A379017
a(n) is the number of distinct sums s(m) + s(m+1) + ... + s(m+n-1), where s = A000002, and m >= 1.
Original entry on oeis.org
2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 3, 4, 3, 4, 5, 4, 5, 6, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 6, 5, 6, 7, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 5, 6, 5, 6, 7, 6, 7, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 7
Offset: 1
Starting with s = (1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, ...) we form a shifted partial sum array:
(row 1) = (1,2,2,1,1,2,1,2,2,...)
(row 2) = (s(1)+s(2), s(2)+s(3), s(3)+s(4), ...) = (3,4,3,2,3,3,3,4,...) = A333229
(row 3) = (s(1)+s(2)+s(3), s(2)+s(3)+s(4), s(3)+s(4)+s(5), ...) = (5,5,4,4,4,5,5,5,5,5,5,4,...)
The number of distinct numbers in (row 3) is 2, so a(3) = 2.
The first 12 rows of the shifted partial sum array: (1, 2), (2, 3, 4), (4, 5), (5, 6, 7), (6, 7, 8, 9), (8, 9, 10), (9, 10, 11, 12), (11, 12, 13), (13, 14), (14, 15, 16), (15, 16, 17, 18), (17, 18, 19). These rows illustrate that fact that the integers in each row are consecutive.
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s = Prepend[Nest[Flatten[Partition[#, 2] /. {{2, 2} -> {2, 2, 1, 1}, {2, 1} -> {2, 2, 1}, {1, 2} -> {2, 1, 1}, {1, 1} -> {2, 1}}] &, {2, 2}, 24], 1]; (* A000002 *)
Length[s]
r[1] = s;
r[n_] := r[n] = Rest[r[n - 1]];
c[n_] := c[n] = Take[r[n], 1000];
sum[n_] := Sum[c[k], {k, 1, n}];
t = Table[Union[sum[n]], {n, 1, 100}]
Map[Length, t]
Showing 1-6 of 6 results.
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