A329864 -id:A329864 - cp's OEIS frontend

A329865 Numbers whose binary expansion has the same runs-resistance as cuts-resistance.

0, 8, 12, 14, 17, 24, 27, 28, 35, 36, 39, 47, 49, 51, 54, 57, 61, 70, 73, 78, 80, 99, 122, 130, 156, 175, 184, 189, 190, 198, 204, 207, 208, 215, 216, 226, 228, 235, 243, 244, 245, 261, 271, 283, 295, 304, 313, 321, 322, 336, 352, 367, 375, 378, 379, 380, 386

Offset: 1

Gus Wiseman, Nov 23 2019

nonn

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The sequence of terms together with their binary expansions begins:
    0:
    8:      1000
   12:      1100
   14:      1110
   17:     10001
   24:     11000
   27:     11011
   28:     11100
   35:    100011
   36:    100100
   39:    100111
   47:    101111
   49:    110001
   51:    110011
   54:    110110
   57:    111001
   61:    111101
   70:   1000110
   73:   1001001
   78:   1001110
   80:   1010000
For example, 36 has runs-resistance 3 because we have (100100) -> (1212) -> (1111) -> (4), while the cuts-resistance is also 3 because we have (100100) -> (00) -> (0) -> ().
Similarly, 57 has runs-resistance 3 because we have (111001) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have (111001) -> (110) -> (1) -> ().

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of 0's in A329867.
The version for runs-resistance equal to cuts-resistance minus 1 is A329866.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Runs-resistance of binary expansion is A318928.
Cuts-resistance of binary expansion is A319416.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Binary words counted by runs-resistance are A319411 and A329767.
Binary words counted by cuts-resistance are A319421 and A329860.
Cf. A000975, A003242, A107907, A164707, A319420, A329738, A329868.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Select[Range[0,100],#==0||runsres[IntegerDigits[#,2]]==degdep[IntegerDigits[#,2]]&]

A329863 Number of compositions of n with cuts-resistance 2.

Original entry on oeis.org

0, 0, 1, 0, 3, 6, 9, 22, 47, 88, 179, 354, 691, 1344, 2617, 5042, 9709, 18632, 35639, 68010, 129556, 246202, 467188, 885036, 1674211, 3163094, 5969022, 11251676, 21189382, 39867970, 74950464, 140798302, 264313039, 495861874, 929709687, 1742193854, 3263069271, 6108762316

Offset: 0

Gus Wiseman, Nov 23 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(2) = 1 through a(7) = 22 compositions (empty column not shown):
  (1,1)  (2,2)    (1,1,3)    (3,3)      (1,1,5)
         (1,1,2)  (1,2,2)    (1,1,4)    (1,3,3)
         (2,1,1)  (2,2,1)    (4,1,1)    (2,2,3)
                  (3,1,1)    (1,1,2,2)  (3,2,2)
                  (1,1,2,1)  (1,1,3,1)  (3,3,1)
                  (1,2,1,1)  (1,2,2,1)  (5,1,1)
                             (1,3,1,1)  (1,1,2,3)
                             (2,1,1,2)  (1,1,3,2)
                             (2,2,1,1)  (1,1,4,1)
                                        (1,4,1,1)
                                        (2,1,1,3)
                                        (2,1,2,2)
                                        (2,2,1,2)
                                        (2,3,1,1)
                                        (3,1,1,2)
                                        (3,2,1,1)
                                        (1,1,2,1,2)
                                        (1,1,2,2,1)
                                        (1,2,1,1,2)
                                        (1,2,2,1,1)
                                        (2,1,1,2,1)
                                        (2,1,2,1,1)

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Column k = 2 of A329861.
Compositions with cuts-resistance 1 are A003242.
Compositions with runs-resistance 2 are A329745.
Numbers whose binary expansion has cuts-resistance 2 are A329862.
Binary words with cuts-resistance 2 are conjectured to be A027383.
Cuts-resistance of binary expansion is A319416.
Binary words counted by cuts-resistance are A319421 and A329860.
Cf. A000975, A003242, A032020, A114901, A240085, A261983, A319420, A329738, A329744, A329864.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],degdep[#]==2&]],{n,0,10}]

Ca(N) = {1/(1-sum(k=1, N, x^k/(1+x^k)))}
A_x(N) = {my(x='x+O('x^N)); concat([0,0],Vec(-1+(1+sum(m=1,N, Ca(N)*x^(2*m)*(Ca(N)-1)/(1+x^m*(2+x^m*(1+Ca(N))))))/(1-sum(m=1,N, Ca(N)*x^(2*m)/(1+x^m*(2+x^m*(1+Ca(N))))))))}
A_x(38) \\ John Tyler Rascoe, Feb 20 2025

G.f.: -1 + (1 + Ca(x) * Sum_{m>0} x^(2*m) * (Ca(x)-1)/(1 + x^m * (2 + x^m * (1+Ca(x)))))/(1 - Ca(x) * Sum_{m>0} x^(2*m)/(1 + x^m * (2 + x^m * (1+Ca(x))))) where Ca(x) is the g.f. for A003242. - John Tyler Rascoe, Feb 20 2025

a(21) onwards from John Tyler Rascoe, Feb 20 2025

A329867 Runs-resistance minus cuts-resistance of the binary expansion of n.

Original entry on oeis.org

0, -1, 1, -1, 1, 1, 1, -2, 0, 1, 1, 2, 0, 2, 0, -3, -1, 0, 3, 2, 2, 1, 3, 1, 0, 2, 2, 0, 0, 1, -1, -4, -2, -1, 2, 0, 0, 3, 2, 0, 1, 3, 1, 2, 1, 2, 2, 0, -1, 0, 1, 0, 2, 2, 0, -1, -1, 0, 1, -1, -1, 0, -2, -5, -3, -2, 1, -1, -1, 2, 0, 1, -1, 0, 3, 4, 2, 3, 0

Offset: 0

Gus Wiseman, Nov 23 2019

sign

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The sequence of binary expansions together with their runs-resistances and cuts-resistances, and their differences, begins:
   0      (): 0 - 0 =  0
   1     (1): 0 - 1 = -1
   2    (10): 2 - 1 =  1
   3    (11): 1 - 2 = -1
   4   (100): 3 - 2 =  1
   5   (101): 2 - 1 =  1
   6   (110): 3 - 2 =  1
   7   (111): 1 - 3 = -2
   8  (1000): 3 - 3 =  0
   9  (1001): 3 - 2 =  1
  10  (1010): 2 - 1 =  1
  11  (1011): 4 - 2 =  2
  12  (1100): 2 - 2 =  0
  13  (1101): 4 - 2 =  2
  14  (1110): 3 - 3 =  0
  15  (1111): 1 - 4 = -3
  16 (10000): 3 - 4 = -1
  17 (10001): 3 - 3 =  0
  18 (10010): 5 - 2 =  3
  19 (10011): 4 - 2 =  2
  20 (10100): 4 - 2 =  2

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of 0's are A329865.
Positions of -1's are A329866.
Sorted positions of first appearances are A329868.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
Runs-resistance of binary expansion is A318928.
Cuts-resistance of binary expansion is A319416.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Binary words counted by runs-resistance are A319411 and A329767.
Binary words counted by cuts-resistance are A319421 and A329860.
Cf. A000975, A003242, A107907, A164707, A329738.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,100}]

For n > 1, a(2^n) = 3 - n.

For n > 1, a(2^n - 1) = 1 - n.

A329866 Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.

Original entry on oeis.org

1, 3, 16, 30, 33, 48, 55, 56, 59, 60, 67, 68, 72, 79, 95, 97, 110, 112, 118, 120, 121, 125, 134, 135, 137, 143, 145, 158, 160, 195, 196, 219, 220, 225, 231, 241, 250, 258, 270, 280, 286, 291, 292, 315, 316, 351, 381, 382, 390, 391, 393, 399, 415, 416, 431, 432

Offset: 1

Gus Wiseman, Nov 23 2019

nonn

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The sequence of terms together with their binary expansions begins:
    1:         1
    3:        11
   16:     10000
   30:     11110
   33:    100001
   48:    110000
   55:    110111
   56:    111000
   59:    111011
   60:    111100
   67:   1000011
   68:   1000100
   72:   1001000
   79:   1001111
   95:   1011111
   97:   1100001
  110:   1101110
  112:   1110000
  118:   1110110
  120:   1111000
For example, 79 has runs-resistance 3 because we have (1001111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have (1001111) -> (0111) -> (11) -> (1) -> (), so 79 is in the sequence.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of -1's in A329867.
The version for runs-resistance equal to cuts-resistance is A329865.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
Runs-resistance of binary expansion is A318928.
Cuts-resistance of binary expansion is A319416.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Binary words counted by runs-resistance are A319411 and A329767.
Binary words counted by cuts-resistance are A319421 and A329860.
Cf. A000975, A003242, A107907, A164707, A329738, A329868.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Select[Range[100],runsres[IntegerDigits[#,2]]-degdep[IntegerDigits[#,2]]==-1&]

A329868 Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.

Original entry on oeis.org

0, 1, 2, 7, 11, 15, 18, 31, 63, 75, 127, 255, 511, 1023, 1234, 2047, 4095, 8191, 9638, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607

Offset: 1

Gus Wiseman, Nov 23 2019

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The sequence of terms together with their binary expansions begins:
      0:
      1:                1
      2:               10
      7:              111
     11:             1011
     15:             1111
     18:            10010
     31:            11111
     63:           111111
     75:          1001011
    127:          1111111
    255:         11111111
    511:        111111111
   1023:       1111111111
   1234:      10011010010
   2047:      11111111111
   4095:     111111111111
   8191:    1111111111111
   9638:   10010110100110
  16383:   11111111111111
  32767:  111111111111111
  65535: 1111111111111111

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Sorted positions of first appearances in A329867.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Runs-resistance of binary expansion is A318928.
Cuts-resistance of binary expansion is A319416.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Binary words counted by runs-resistance are A319411 and A329767.
Binary words counted by cuts-resistance are A319421 and A329860.
Cf. A000975, A003242, A107907, A164707, A329738, A329865, A329866.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
das=Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,1000000}];
Table[Position[das,i][[1,1]]-1,{i,First/@Gather[das]}]

A329869 Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 4, 5, 11, 19, 36, 77, 138, 252, 528, 1072, 2204, 4634, 9575, 19732, 40754

Offset: 0

Gus Wiseman, Nov 23 2019

A composition of n is a finite sequence of positive integers summing to n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(1) = 1 through a(9) = 19 compositions:
  1   2   3   4   5   6      7       8        9
      11      22      33     11113   44       11115
                      11112  31111   11114    12222
                      21111  111211  41111    22221
                             112111  111122   51111
                                     111311   111222
                                     113111   111411
                                     211112   114111
                                     221111   211113
                                     1111121  222111
                                     1211111  311112
                                              1111131
                                              1111221
                                              1112112
                                              1121112
                                              1221111
                                              1311111
                                              2111211
                                              2112111
For example, the runs-resistance of (1221111) is 3 because we have: (1221111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have: (1221111) -> (2111) -> (11) -> (1) -> (), so (1221111) is counted under a(9).

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

The version for binary indices is A329866.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329864, A329865, A329867, A329868.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]+1==degdep[#]&]],{n,0,10}]

A330028 Number of compositions of n with cuts-resistance <= 2.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 23, 45, 86, 159, 303, 568, 1069, 2005, 3769, 7066, 13251, 24821, 46482, 86988, 162758

Offset: 0

Gus Wiseman, Nov 27 2019

A composition of n is a finite sequence of positive integers summing to n.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)    (3)    (4)      (5)
           (1,1)  (1,2)  (1,3)    (1,4)
                  (2,1)  (2,2)    (2,3)
                         (3,1)    (3,2)
                         (1,1,2)  (4,1)
                         (1,2,1)  (1,1,3)
                         (2,1,1)  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,2,1)
                                  (1,2,1,1)

Sum of first three columns of A329861.
Compositions with cuts-resistance 1 are A003242.
Compositions with cuts-resistance 2 are A329863.
Compositions with runs-resistance 2 are A329745.
Numbers whose binary expansion has cuts-resistance 2 are A329862.
Binary words with cuts-resistance 2 are A027383.
Cuts-resistance of binary expansion is A319416.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A000975, A003242, A032020, A114901, A240085, A261983, A319420, A329738, A329744, A329864.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],degdep[#]<=2&]],{n,0,10}]

cp's OEIS Frontend

A329865 Numbers whose binary expansion has the same runs-resistance as cuts-resistance.

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A329863 Number of compositions of n with cuts-resistance 2.

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A329867 Runs-resistance minus cuts-resistance of the binary expansion of n.

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A329866 Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.

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A329868 Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.

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A329869 Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.

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A330028 Number of compositions of n with cuts-resistance <= 2.

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