A319411 -id:A319411 - cp's OEIS frontend

A319413 Number of trailing zero entries in row n of triangle A319411.

0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 57

Offset: 1

N. J. A. Sloane, Sep 20 2018

nonn

See triangle in A319411.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..75
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.

Cf. A318928, A319411.

a(21)-a(69) from Hiroaki Yamanouchi, Sep 25 2018

A318928 Runs-resistance of binary representation of n.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 3, 3, 2, 4, 2, 4, 3, 1, 3, 3, 5, 4, 4, 2, 5, 4, 3, 4, 4, 3, 3, 4, 3, 1, 3, 3, 5, 3, 3, 5, 4, 3, 4, 5, 2, 4, 3, 4, 5, 4, 3, 3, 3, 2, 4, 4, 3, 3, 2, 3, 4, 3, 3, 4, 3, 1, 3, 3, 5, 3, 3, 5, 3, 4, 3, 3, 5, 6, 4, 5, 3, 3, 4, 5, 4, 4, 4, 2, 5, 4, 5, 5, 4, 5, 5, 4, 5, 4

Offset: 1

N. J. A. Sloane, Sep 09 2018

Following Lenormand (2003), we define the "runs-resistance" of a finite list L to be the number of times the RUNS transformation must be applied to L in order to reduce L to a list with a single element.
Here it is immaterial whether we read the binary representation of n from left to right or right to left.
The RUNS transformation must be applied at least once, in order to obtain a list, so a(n) >= 1.

			11 in binary is [1, 0, 1, 1],
which has runs of lengths [1, 1, 2],
which has runs of lengths [2, 1],
which has runs of lengths [1, 1],
which has a single run of length [2].
This took four steps, so a(11) = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.
N. J. A. Sloane, Transforms

See A319103 for an inverse, and A319417 and A319418 for records.
Ignoring the first digit gives A329870.
Cuts-resistance is A319416.
Compositions counted by runs-resistance are A329744.
Binary words counted by runs-resistance are A319411 and A329767.
Cf. A107907, A319420, A329860, A329861, A329865, A329867.

with(transforms);
# compute Lenormand's "resistance" of a list
resist:=proc(a) local ct,i,b;
if whattype(a) <> list then ERROR("input must be a list"); fi:
ct:=0; b:=a; for i from 1 to 100 do
if nops(b)=1 then return(ct); fi;
b:=RUNS(b); ct:=ct+1; od; end;
a:=[1];
for n from 2 to 100 do
b:=convert(n,base,2);
r:=resist(b);
a:=[op(a),r];
od:

Table[If[n == 1, 1, Length[NestWhileList[Length/@Split[#] &, IntegerDigits[n, 2], Length[#] > 1 &]] - 1], {n, 50}] (* Gus Wiseman, Nov 25 2019 *)

a(1) corrected by N. J. A. Sloane, Sep 20 2018

A319416 Cuts-resistance of n: number of applications of Lernormand's "raboter" map needed to transform the binary expansion of n to the empty string.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2, 3, 3, 2, 2, 3, 3, 3, 4, 5, 5, 4, 3, 3, 3, 2, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 4, 3, 2, 2, 2, 2, 3, 4, 3, 3, 3, 4, 4, 4, 5, 6, 6, 5, 4, 4, 4, 3, 3, 3, 4, 3, 2, 2, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2

Offset: 0

N. J. A. Sloane, Sep 21 2018

Here we are using Lenormand's "raboter" map in a stricter sense than in A318921 and A319419. If S is a binary string with successive runs of lengths b,c,d,e,..., the "raboter" map sends S to the binary string with successive runs of lengths b-1,c-1,d-1,e-1,... Runs of length 0 are omitted (they are indicated by dots in the examples below).
To get a(n), start with S equal to the binary expansion of n beginning with the most significant bit, and keep applying the map until we reach the empty string.
After the first step, the string may start with a string of 0's: this is acceptable because we are working with strings, not binary expansions of numbers.
For example, 34 = 100010 -> .00.. = 00 -> 0. = 0 -> . (the empty string), taking 3 steps, so a(34) = 3.
Note: this is not the same as the number of applications of the map k -> A318921(k) needed to reduce the binary expansion of n to zero (because A318921 does not distinguish between 0 and the empty string).
This is also not the same as the number of applications of the map k -> A319419(k) needed to reduce the binary expansion of n to -1 (because A319419 does not distinguish between a string of 0's and a single 0).
The value k appears for the first time when n = 2^k - 1.

			n: repeatedly applying the map / number of steps = a(n)
0: 0 -> . / 1
1: 1 -> . / 1
2: 10 -> . / 1
3: 11 -> 1 -> . / 2
4: 100 -> 0 -> . / 2
5: 101 -> . / 1
6: 110 -> 1 -> . / 2
7: 111 -> 11 -> 1 -> . / 3
8: 1000 -> 00 -> 0 -> . / 3
9: 1001 -> 0 -> . / 2
10: 1010 -> . / 1
11: 1011 -> 1 -> . / 2
12: 1100 -> 10 -> . / 2
...

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Positions of 1's are A000975.
Positions of 2's are A329862.
The version for runs-resistance is A318928.
The version for compositions is A329861.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A318921, A319411, A319419, A319420, A329744, A329767, A329863, A329865.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[degdep[IntegerDigits[n,2]],{n,0,50}] (* Gus Wiseman, Nov 25 2019 *)

a(n) = my (b=binary(n), w=#b); for (k=1, oo, my (ww=0); for (i=2, w, if (b[i-1]==b[i], b[ww++]=b[i])); if (ww==0, return (k), w=ww)) \\ Rémy Sigrist, Sep 23 2018

More terms from Rémy Sigrist, Sep 23 2018

A329767 Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n.

Original entry on oeis.org

1, 2, 0, 0, 2, 2, 0, 2, 2, 4, 0, 2, 4, 6, 4, 0, 2, 2, 12, 12, 4, 0, 2, 6, 30, 18, 8, 0, 0, 2, 2, 44, 44, 32, 4, 0, 0, 2, 6, 82, 76, 74, 16, 0, 0, 0, 2, 4, 144, 138, 172, 52, 0, 0, 0, 0, 2, 6, 258, 248, 350, 156, 4, 0, 0, 0, 0, 2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Nov 21 2019

A composition of n is a finite sequence of positive integers with sum n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
Except for the k = 0 column and the n = 0 and n = 1 rows, this is the triangle appearing on page 3 of Lenormand, which is A319411. Unlike A318928, we do not here require that a(n) >= 1.
The n = 0 row is chosen to ensure that the row-sums are A000079, although the empty word arguably has indeterminate runs-resistance.

			Triangle begins:
   1
   2   0
   0   2   2
   0   2   2   4
   0   2   4   6   4
   0   2   2  12  12   4
   0   2   6  30  18   8   0
   0   2   2  44  44  32   4   0
   0   2   6  82  76  74  16   0   0
   0   2   4 144 138 172  52   0   0   0
   0   2   6 258 248 350 156   4   0   0   0
   0   2   2 426 452 734 404  28   0   0   0   0
For example, row n = 4 counts the following words:
  0000  0011  0001  0010
  1111  0101  0110  0100
        1010  0111  1011
        1100  1000  1101
              1001
              1110

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

Row sums are A000079.
Column k = 2 is A319410.
Column k = 3 is 2 * A329745.
The version for compositions is A329744.
The version for partitions is A329746.
The number of nonzero entries in row n > 0 is A319412(n).
The runs-resistance of the binary expansion of n is A318928.
Cf. A001037, A096365, A225485, A245563, A319411, A325280, A329747, A329750.

runsres[q_]:=If[Length[q]==1,0,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
Table[Length[Select[Tuples[{0,1},n],runsres[#]==k&]],{n,0,10},{k,0,n}]

A329860 Triangle read by rows where T(n,k) is the number of binary words of length n with cuts-resistance k.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 2, 4, 2, 0, 2, 8, 4, 2, 0, 2, 12, 12, 4, 2, 0, 2, 20, 22, 14, 4, 2, 0, 2, 28, 48, 28, 16, 4, 2, 0, 2, 44, 84, 70, 32, 18, 4, 2, 0, 2, 60, 162, 136, 90, 36, 20, 4, 2, 0, 2, 92, 276, 298, 178, 110, 40, 22, 4, 2, 0, 2, 124, 500, 564, 432, 220, 132, 44, 24, 4, 2

Offset: 0

Gus Wiseman, Nov 23 2019

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.

			Triangle begins:
   1
   0   2
   0   2   2
   0   2   4   2
   0   2   8   4   2
   0   2  12  12   4   2
   0   2  20  22  14   4   2
   0   2  28  48  28  16   4   2
   0   2  44  84  70  32  18   4   2
   0   2  60 162 136  90  36  20   4   2
   0   2  92 276 298 178 110  40  22   4   2
   0   2 124 500 564 432 220 132  44  24   4   2
Row n = 4 counts the following words:
  0101  0010  0001  0000
  1010  0011  0111  1111
        0100  1000
        0110  1110
        1001
        1011
        1100
        1101

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

Column k = 2 appears to be 2 * A027383.
The version for runs-resistance is A319411 or A329767.
The cuts-resistance of the binary expansion of n is A319416(n).
The version for compositions is A329861.
Cf. A000975, A164707, A261983, A318928, A319420, A319421, A329738, A329865.

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

For positive indices, T(n,k) = 2 * A319421(n,k).

A329861 Triangle read by rows where T(n,k) is the number of compositions of n with cuts-resistance k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 3, 0, 1, 0, 7, 6, 2, 0, 1, 0, 14, 9, 6, 2, 0, 1, 0, 23, 22, 10, 6, 2, 0, 1, 0, 39, 47, 22, 10, 7, 2, 0, 1, 0, 71, 88, 52, 24, 10, 8, 2, 0, 1, 0, 124, 179, 101, 59, 26, 11, 9, 2, 0, 1, 0, 214, 354, 220, 112, 71, 28, 12, 10, 2, 0, 1

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

			Triangle begins:
  1
  0  1
  0  1  1
  0  3  0  1
  0  4  3  0  1
  0  7  6  2  0  1
  0 14  9  6  2  0  1
  0 23 22 10  6  2  0  1
  0 39 47 22 10  7  2  0  1
  0 71 88 52 24 10  8  2  0  1
Row n = 6 counts the following compositions (empty columns not shown):
  (6)     (33)    (222)    (11112)  (111111)
  (15)    (114)   (1113)   (21111)
  (24)    (411)   (3111)
  (42)    (1122)  (11121)
  (51)    (1131)  (11211)
  (123)   (1221)  (12111)
  (132)   (1311)
  (141)   (2112)
  (213)   (2211)
  (231)
  (312)
  (321)
  (1212)
  (2121)

Row sums are A000079.
Column k = 1 is A003242 (for n > 0).
Column k = 2 is A329863.
Row sums without the k = 1 column are A261983.
The version for runs-resistance is A329744.
The version for binary vectors is A329860.
The cuts-resistance of the binary expansion of n is A319416.
Cf. A175342, A240085, A319411, A319420, A319421, A329738, A329862, A329865.

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

A319421 Triangle read by rows: T(n,k) (1 <= k <= n) = one-half of the number of binary vectors of length n and cuts-resistance k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 6, 2, 1, 1, 10, 11, 7, 2, 1, 1, 14, 24, 14, 8, 2, 1, 1, 22, 42, 35, 16, 9, 2, 1, 1, 30, 81, 68, 45, 18, 10, 2, 1, 1, 46, 138, 149, 89, 55, 20, 11, 2, 1, 1, 62, 250, 282, 216, 110, 66, 22, 12, 2, 1, 1, 94, 419, 577, 422, 285, 132, 78, 24, 13, 2, 1

Offset: 1

N. J. A. Sloane, Sep 23 2018

Cuts-resistance is defined in A319416.
This triangle summarizes the data shown in A319420.
Conjecture (Sloane): Sum_{i = 1..n} i * T(n,i) = A189391(n).

			Triangle begins:
   1
   1   1
   1   2   1
   1   4   2   1
   1   6   6   2   1
   1  10  11   7   2   1
   1  14  24  14   8   2   1
   1  22  42  35  16   9   2   1
   1  30  81  68  45  18  10   2   1
   1  46 138 149  89  55  20  11   2   1
   1  62 250 282 216 110  66  22  12   2   1
   1  94 419 577 422 285 132  78  24  13   2   1
Lenormand gives first 15 rows.
For example, the "1,2,1" row here refers to the 8 vectors of length 3. There are 2 vectors of cuts-resistance 1, namely 010 and 101 (see A319416), 4 vectors of cuts-resistance 2 (100,011,001,110), and 2 of cuts-resistance 3 (000 and 111). Halving these counts we get 1,2,1

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. See page 4.
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Row sums are A000079.
Column k = 2 appears to be A027383.
The version for runs-resistance is A319411 or A329767.
The version for compositions is A329861.
The cuts-resistance of the binary expansion of n is A319416(n).
Cf. A000975, A164707, A189391, A261983, A318921, A318928, A319420, A329738, A329860, A329865.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Tuples[{0,1},n],First[#]==1&°dep[#]==k&]],{n,8},{k,n}] (* Gus Wiseman, Nov 25 2019 *)

T(n,k) = A329860(n,k)/2. - Gus Wiseman, Nov 25 2019

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

Original entry on oeis.org

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]]&]

A319420 Irregular triangle read by rows: row n lists the cuts-resistances of the 2^n binary vectors of length n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 22 2018

The cuts-resistance of a vector is defined in A319416. The 2^n vectors of length n are taken in lexicographic order.
Note that here the vectors can begin with either 0 or 1, whereas in A319416 only vectors beginning with 1 are considered (since there we are considering binary representations of numbers).
Conjecture: The row sums, halved, appear to match A189391.

			Triangle begins:
0,
1,1,
2,1,1,2,
3,2,1,2,2,1,2,3,
4,3,2,2,2,1,2,3,3,2,1,2,2,2,3,4,
5,4,3,3,3,2,2,3,3,2,1,2,2,2,3,4,4,3,2,2,2,1,2,3,3,2,2,2,3,3,3,4,5,
...

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. See table on page 4.

Keeping the first digit gives A319416.
Positions of 1's are the terms > 1 of A061547 and A086893, all minus 1.
The version for runs-resistance is A329870.
Compositions counted by cuts-resistance are A329861.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A000975, A027383, A189391, A318921, A318928, A319411, A329767, A329862, A329865.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[degdep[Rest[IntegerDigits[n,2]]],{n,0,50}] (* Gus Wiseman, Nov 25 2019 *)

A329862 Positive integers whose binary expansion has cuts-resistance 2.

Original entry on oeis.org

3, 4, 6, 9, 11, 12, 13, 18, 19, 20, 22, 25, 26, 37, 38, 41, 43, 44, 45, 50, 51, 52, 53, 74, 75, 76, 77, 82, 83, 84, 86, 89, 90, 101, 102, 105, 106, 149, 150, 153, 154, 165, 166, 169, 171, 172, 173, 178, 179, 180, 181, 202, 203, 204, 205, 210, 211, 212, 213

Offset: 1

Gus Wiseman, Nov 23 2019

nonn

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:
   3:      11
   4:     100
   6:     110
   9:    1001
  11:    1011
  12:    1100
  13:    1101
  18:   10010
  19:   10011
  20:   10100
  22:   10110
  25:   11001
  26:   11010
  37:  100101
  38:  100110
  41:  101001
  43:  101011
  44:  101100
  45:  101101
  50:  110010

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

Positions of 2's in A319416.
Numbers whose binary expansion has cuts-resistance 1 are A000975.
Binary words with cuts-resistance 2 are conjectured to be A027383.
Compositions with cuts-resistance 2 are A329863.
Cuts-resistance of binary expansion without first digit is A319420.
Binary words counted by cuts-resistance are A319421 and A329860.
Compositions counted by cuts-resistance are A329861.
Cf. A107907, A114901, A164707, A318928, A319411, A329745, A329865, A329866.

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

cp's OEIS Frontend

A319413 Number of trailing zero entries in row n of triangle A319411.

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A318928 Runs-resistance of binary representation of n.

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A319416 Cuts-resistance of n: number of applications of Lernormand's "raboter" map needed to transform the binary expansion of n to the empty string.

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A329767 Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n.

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A329860 Triangle read by rows where T(n,k) is the number of binary words of length n with cuts-resistance k.

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A329861 Triangle read by rows where T(n,k) is the number of compositions of n with cuts-resistance k.

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A319421 Triangle read by rows: T(n,k) (1 <= k <= n) = one-half of the number of binary vectors of length n and cuts-resistance k.

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

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