A054353 -id:A054353 - cp's OEIS frontend

A329360 The decimal expansion of a(n) is the first n terms of A000002.

0, 1, 12, 122, 1221, 12211, 122112, 1221121, 12211212, 122112122, 1221121221, 12211212212, 122112122122, 1221121221221, 12211212212211, 122112122122112, 1221121221221121, 12211212212211211, 122112122122112112, 1221121221221121122, 12211212212211211221

Offset: 0

Gus Wiseman, Nov 12 2019

Cf. A000002, A054353, A088568, A288605, A296658, A329315, A329316, A329355, A329356, A329361, A329362.

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=If[n==0,{},Nest[kolagrow,{1},n-1]];
Table[FromDigits[kol[n]],{n,0,30}]

A329361 a(n) = Sum_{i = 1..n} 2^(n - i) * A000002(i).

Original entry on oeis.org

0, 1, 4, 10, 21, 43, 88, 177, 356, 714, 1429, 2860, 5722, 11445, 22891, 45784, 91569, 183139, 366280, 732562, 1465125, 2930252, 5860505, 11721011, 23442024, 46884049, 93768100, 187536202, 375072405, 750144811, 1500289624, 3000579249, 6001158499, 12002317000

Offset: 0

Gus Wiseman, Nov 12 2019

nonn

			The first 5 terms of A000002 are {1, 2, 2, 1, 1}, so a(5) = 2^4 * 1 + 2^3 * 2 + 2^2 * 2 + 2^1 * 1 + 2^0 * 1 = 43.

Cf. A000002, A054353, A088568, A288605, A296658, A329315, A329316, A329355, A329356, A329360, A329362.

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=If[n==0,{},Nest[kolagrow,{1},n-1]];
Table[FromDigits[kol[n],2],{n,0,30}]

a(n + 1) = A000002(n) + 2 a(n).

A333229 First sums of the Kolakoski sequence A000002.

Original entry on oeis.org

3, 4, 3, 2, 3, 3, 3, 4, 3, 3, 4, 3, 2, 3, 3, 2, 3, 4, 3, 3, 3, 2, 3, 3, 3, 4, 3, 2, 3, 3, 2, 3, 3, 3, 4, 3, 3, 4, 3, 2, 3, 3, 3, 4, 3, 3, 3, 2, 3, 3, 2, 3, 4, 3, 3, 4, 3, 2, 3, 3, 3, 4, 3, 3, 4, 3, 2, 3, 3, 2, 3, 3, 3, 4, 3, 3, 3, 2, 3, 4, 3, 3, 4, 3, 2, 3, 3

Offset: 1

Gus Wiseman, Mar 18 2020

nonn

Positions of 3's are A054353.
Positions of 2's are A074262.
Positions of 4's are A074263.
The number of runs in the first n terms of A000002 is A156253(n).
Even-indexed terms are A332273 (without the first term).
Odd-indexed terms are A332875.
Cf. A000002, A001462, A013947, A013948, A088568, A156728, A288605, A296658, A329315, A329316, A329317, A329362.

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=Nest[kolagrow,{1},n-1];
Table[kol[n][[-1]]+kol[n+1][[-1]],{n,30}]

a(n) = A000002(n) + A000002(n + 1).

A329758 Number of n-th generation nodes of a rooted binary tree whose m-th node has exactly A000002(m) descendants, where A000002 is the Kolakoski sequence.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 15, 22, 34, 51, 76, 114, 171, 257, 385, 575, 862, 1295, 1941, 2916, 4374, 6558, 9840, 14766, 22151, 33236, 49864, 74775, 112144, 168198, 252313, 378489, 567747, 851649, 1277446, 1916182, 2874172, 4311325, 6466984, 9700248, 14550387, 21825590

Offset: 1

Jakub Zaborowski, Nov 20 2019

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..62

Cf. A000002, A054352, A054353.

def A329758():
   x = 1
   g = A000002()
   while True:
       yield x
       acc = 0
       for i in range(0, x):
           acc = acc + next(g)
       x = acc # Jack W Grahl, May 04 2020

a(n) = A054353(a(1) + ... + a(n-1)) - A054353(a(1) + ... + a(n-2)) for n > 2.

a(n) = A054352(n-1) - A054352(n-2). - A.H.M. Smeets, Apr 08 2024

More terms from Jack W Grahl, May 04 2020

A229785 Partial sums of A157129.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 9, 10, 12, 14, 16, 18, 19, 20, 22, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 38, 40, 42, 43, 44, 45, 46, 48, 50, 52, 54, 55, 56, 58, 60, 61, 62, 63, 64, 66, 68, 70, 72, 73, 74, 76, 78, 79, 80, 81, 82, 84, 86, 88, 90, 91, 92, 94, 96, 97, 98, 100, 102, 103, 104

Offset: 1

Benoit Cloitre, Sep 29 2013

nonn

Although the behavior of the partial sums of the Kolakoski sequence (A054353) is mysterious, this sequence is much easier to handle.

Cf. A054353, A157129.

a(n)=(3/2)n+O(1). More precisely, let b(n)=3*n-2*a(n); then b(n) satisfies the following recurrence modulo 12: b(n)=1,2,1,0,1,2,3,4,3,2,1 for n=1,2,3,4,5,6,7,8,9,10,11. Then for k>=1 we have b(12k)=b(4k), b(12k+1)=b(4k+1), b(12k+2)=b(4k+2), b(12k+2)=b(4k+2), b(12k+3)=b(4k+2)-1, b(12k+4)=b(4k+2)-2, b(12k+5)=b(4k+2)-1, b(12k+6)=b(4k+2), b(12k+7)=4-b(4k+3), b(12k+8)=4-b(4k+4), b(12k+9)=4-b(4k+3), b(12k+10)=4-b(4k+2), b(12k+11)=b(4k+3).

A216348 Numbers that appear in either both A156242(n) + 1 and A156243(n) or both A156242(n) and A156243(n) + 1.

Original entry on oeis.org

6, 7, 10, 15, 20, 21, 24, 25, 30, 33, 34, 37, 42, 43, 46, 47, 50, 55, 60, 61, 64, 69, 72, 73, 76, 77, 82, 87, 88, 91, 96, 101, 102, 105, 106, 109, 114, 117, 118, 123, 128, 129, 132, 137, 142, 143, 146, 147

Offset: 1

Jon Perry, Sep 04 2012

nonn

			6 is in both A156242 and A156243 + 1.
7 is in both A156242 + 1 and A156243.

Cf. A156242, A156243, A054353.

n = 10; t = 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}, n], 1]; t2 = Accumulate[t]; {t3, t4} = Transpose[Partition[t2, 2]]; Union[Intersection[t3, t4 + 1], Intersection[t3 + 1, t4]] (* T. D. Noe, Sep 26 2012 *)

A289324 Number of twos minus number of ones in the first 10^n entries of the Kolakoski sequence, A000002.

Original entry on oeis.org

-1, 0, 2, -4, 8, 56, 28, -92, -1350, -2446, 4658, -3174, -101402, -16318, -632474, -1954842, 10724544, 45041304, 111069790, 548593100, 1818298480

Offset: 0

Richard P. Brent, Jul 07 2017

This is equivalent to A195206, since a(n) = (#twos)-(#ones) = 10^n-2*(#ones) in the first 10^n entries of A000002.
For example, a(2) = 51 - 49 = (100 - 49) - 49 = 100 - 2*49 = 2 because there are 49 ones and 51 twos in the first 100 = 10^2 entries of A000002.
The entries in this sequence appear to be of order 10^(n/2), whereas the entries in A195206 are larger (of order 10^n).
This sequence is analogous to A289323; the difference is that the indices are powers of ten instead of powers of two.

			The first 10 entries in the Kolakoski sequence, A000002, are 1221121221. There are 5 ones and 5 twos, so a(1) = 5 - 5 = 0.
The first 100=10^2 entries in the Kolakoski sequence A000002 include 49 ones and 51 twos, so a(2) = 51 - 49 = 2.

See the references and links for A195206, A289322.

Cf. A000002, A195206, A054353, A074286, A088568, A156077.

a(n) = 10^n - 2*A195206(n).

Additional (20th) term from Richard P. Brent, Mar 01 2018

cp's OEIS Frontend

A329360 The decimal expansion of a(n) is the first n terms of A000002.

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A329361 a(n) = Sum_{i = 1..n} 2^(n - i) * A000002(i).

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A333229 First sums of the Kolakoski sequence A000002.

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A329758 Number of n-th generation nodes of a rooted binary tree whose m-th node has exactly A000002(m) descendants, where A000002 is the Kolakoski sequence.

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A229785 Partial sums of A157129.

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A216348 Numbers that appear in either both A156242(n) + 1 and A156243(n) or both A156242(n) and A156243(n) + 1.

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A289324 Number of twos minus number of ones in the first 10^n entries of the Kolakoski sequence, A000002.

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