cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A329359 Irregular triangle read by rows where row n gives the lengths of the factors in the co-Lyndon factorization of the binary expansion of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 4, 3, 1, 4, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 5, 2, 2, 1, 2, 3, 2, 1, 1, 1, 5, 4, 1, 5, 3, 1, 1, 5, 4, 1, 5, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 1, 3, 3, 3, 2, 1, 3, 3, 3, 1, 1, 1, 6, 5, 1, 2, 2, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Nov 12 2019

Keywords

Comments

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

Examples

			Triangle begins:
   1: (1)       21: (221)      41: (51)       61: (51)
   2: (2)       22: (23)       42: (222)      62: (6)
   3: (11)      23: (2111)     43: (2211)     63: (111111)
   4: (3)       24: (5)        44: (24)       64: (7)
   5: (21)      25: (41)       45: (231)      65: (61)
   6: (3)       26: (5)        46: (24)       66: (52)
   7: (111)     27: (311)      47: (21111)    67: (511)
   8: (4)       28: (5)        48: (6)        68: (43)
   9: (31)      29: (41)       49: (51)       69: (421)
  10: (22)      30: (5)        50: (6)        70: (43)
  11: (211)     31: (11111)    51: (411)      71: (4111)
  12: (4)       32: (6)        52: (6)        72: (7)
  13: (31)      33: (51)       53: (51)       73: (331)
  14: (4)       34: (42)       54: (33)       74: (322)
  15: (1111)    35: (411)      55: (3111)     75: (3211)
  16: (5)       36: (33)       56: (6)        76: (34)
  17: (41)      37: (321)      57: (51)       77: (331)
  18: (32)      38: (33)       58: (6)        78: (34)
  19: (311)     39: (3111)     59: (411)      79: (31111)
  20: (5)       40: (6)        60: (6)        80: (7)
For example, 45 has binary expansion (101101), with co-Lyndon factorization (10)(110)(1), so row n = 45 is (2,3,1).

Crossrefs

Row lengths are A329312.
Row sums are A070939.
Positions of rows of length 1 are A275692.
The non-"co" version is A329314.
Binary co-Lyndon words are counted by A001037 and ranked by A329318.
Cf. A059966, A211097, A211100, A328596, A296372, A329313, A329315, A329325, A329357.

Programs

  • Mathematica
    colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[Length/@colynfac[If[n==0,{},IntegerDigits[n,2]]],{n,30}]

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Nov 12 2019

Keywords

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Nov 12 2019

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Gus Wiseman, Mar 18 2020

Keywords

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

a(n) = A000002(n) + A000002(n + 1).
Previous Showing 11-14 of 14 results.

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