A328592 -id:A328592 - cp's OEIS frontend

A351200 Number of patterns of length n with all distinct runs.

1, 1, 3, 11, 53, 305, 2051, 15731, 135697, 1300869, 13726431, 158137851, 1975599321, 26607158781, 384347911211, 5928465081703, 97262304328573, 1691274884085061, 31073791192091251, 601539400910369671, 12238270940611270161, 261071590963047040241

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

			The a(1) = 1 through a(3) = 11 patterns:
  (1)  (1,1)  (1,1,1)
       (1,2)  (1,1,2)
       (2,1)  (1,2,2)
              (1,2,3)
              (1,3,2)
              (2,1,1)
              (2,1,3)
              (2,2,1)
              (2,3,1)
              (3,1,2)
              (3,2,1)
The complement for n = 3 counts the two patterns (1,2,1) and (2,1,2).

Andrew Howroyd, Table of n, a(n) for n = 0..200
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A351292.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns, complement A069321.
A005811 counts runs in binary expansion.
A032011 counts patterns with distinct multiplicities.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A060223 counts Lyndon patterns, necklaces A019536, aperiodic A296975.
A131689 counts patterns by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A297770 counts distinct runs in binary expansion.
A345194 counts alternating patterns, up/down A350354.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351202 = permutations of prime factors.
- A351642 = word structures.
Row sums of A351640.
Cf. A003242, A098504, A098859, A106356, A242882, A325545, A328592, A329740, A351014, A351204, A351291.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],UnsameQ@@Split[#]&]],{n,0,6}]

\\ here LahI is A111596 as row polynomials.
LahI(n,y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p,i,y)*LahI(i,y))}
R(q)={[subst(serlaplace(p), y, 1) | p<-Vec(q)]}
seq(n)={my(q=S(n)); concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 12 2022

Terms a(10) and beyond from Andrew Howroyd, Feb 12 2022

A351290 Numbers k such that the k-th composition in standard order has all distinct runs.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78

Offset: 1

Gus Wiseman, Feb 10 2022

nonn

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
   0:      0  ()
   1:      1  (1)
   2:     10  (2)
   3:     11  (1,1)
   4:    100  (3)
   5:    101  (2,1)
   6:    110  (1,2)
   7:    111  (1,1,1)
   8:   1000  (4)
   9:   1001  (3,1)
  10:   1010  (2,2)
  11:   1011  (2,1,1)
  12:   1100  (1,3)
  14:   1110  (1,1,2)
  15:   1111  (1,1,1,1)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for Heinz numbers and prime multiplicities is A130091.
The version using binary expansions is A175413, complement A351205.
The version for run-lengths instead of runs is A329739.
These compositions are counted by A351013.
The complement is A351291.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions:
- Length is A000120.
- Parts are A066099, reverse A228351.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Split[stc[#]]&]

A351292 Number of patterns of length n with all distinct run-lengths.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 57, 61, 109, 161, 1265, 1317, 2469, 3577, 5785, 43901, 47165, 86337, 127665, 204853, 284197, 2280089, 2398505, 4469373, 6543453, 10570993, 14601745, 22502549, 159506453, 171281529, 314077353, 462623821, 742191037, 1031307185, 1580543969, 2141246229

Offset: 0

Gus Wiseman, Feb 10 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

			The a(1) = 1 through a(5) = 9 patterns:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
              (1,1,2)  (1,1,1,2)  (1,1,1,1,2)
              (1,2,2)  (1,2,2,2)  (1,1,1,2,2)
              (2,1,1)  (2,1,1,1)  (1,1,2,2,2)
              (2,2,1)  (2,2,2,1)  (1,2,2,2,2)
                                  (2,1,1,1,1)
                                  (2,2,1,1,1)
                                  (2,2,2,1,1)
                                  (2,2,2,2,1)
The a(6) = 57 patterns grouped by sum:
  111111  111112  111122  112221  111223  111233  112333  122333
          111211  111221  122211  111322  111332  113332  133322
          112111  122111  211122  112222  112223  122233  221333
          211111  221111  221112  211222  113222  133222  223331
                                  221113  122222  211333  333122
                                  222112  211133  222133  333221
                                  222211  221222  222331
                                  223111  222113  233311
                                  311122  222122  331222
                                  322111  222221  332221
                                          222311  333112
                                          233111  333211
                                          311222
                                          322211
                                          331112
                                          332111

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for runs instead of run-lengths is A351200.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns, complement A069321.
A005811 counts runs in binary expansion.
A032011 counts patterns with distinct multiplicities.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A060223 counts Lyndon patterns, necklaces A019536, aperiodic A296975.
A131689 counts patterns by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A165413 counts distinct run-lengths in binary expansion, runs A297770.
A345194 counts alternating patterns, up/down A350354.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351202 = permutations of prime factors.
- A351638 = word structures.
Row sums of A350824.
Cf. A003242, A098504, A098859, A106356, A239455, A242882, A325545, A328592, A329740, A351014, A351293.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u,k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
seq(n)={my(u=P(n), c=poldegree(u[#u])); concat([1], sum(k=1, c, R(u, k)*sum(r=k, c, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 11 2022

From Andrew Howroyd, Feb 12 2022: (Start)
a(n) = Sum_{k=1..n} R(n,k)*(Sum_{r=k..n} binomial(r, k)*(-1)^(r-k)), where R(n,k) = Sum_{j=1..floor((sqrt(8*n+1)-1)/2)} k*(k-1)^(j-1) * j! * A008289(n,j).
G.f.: 1 + Sum_{r>=1} Sum_{k=1..r} R(k,x) * binomial(r, k)*(-1)^(r-k), where R(k,x) = Sum_{j>=1} k*(k-1)^(j-1) * j! * [y^j](Product_{k>=1} 1 + y*x^k).
(End)

Terms a(10) and beyond from Andrew Howroyd, Feb 11 2022

A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 20, 51, 121, 276, 612, 1335, 2881, 6144, 12950, 27029, 55977, 115222, 236058, 481683, 979443

Offset: 0

Gus Wiseman, Jun 16 2025

			The subset {1,3,4,8,9} has maximal runs ((1),(3,4),(8,9)), with lengths (1,2,2), so is counted under a(10).
The a(0) = 0 through a(6) = 20 subsets:
  .  .  .  {1,3}  {1,3}  {1,3}      {1,3}
                  {1,4}  {1,4}      {1,4}
                  {2,4}  {1,5}      {1,5}
                         {2,4}      {1,6}
                         {2,5}      {2,4}
                         {3,5}      {2,5}
                         {1,3,5}    {2,6}
                         {1,2,4,5}  {3,5}
                                    {3,6}
                                    {4,6}
                                    {1,3,5}
                                    {1,3,6}
                                    {1,4,6}
                                    {2,4,6}
                                    {1,2,4,5}
                                    {1,2,4,6}
                                    {1,2,5,6}
                                    {1,3,4,6}
                                    {1,3,5,6}
                                    {2,3,5,6}

For equal instead of distinct lengths the complement is A243815.
These subsets are ranked by the non-members of A328592.
The complement is counted by A384175.
For strict partitions instead of subsets see A384178, A384884, A384886, A384880.
For permutations instead of subsets see A384891, A384892, A010027.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A044813, A242882, A325325, A329739, A351202, A383013, A384177, A384889, A384890.

Table[Length[Select[Subsets[Range[n]],!UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 18, 27, 46, 77, 122, 191, 326, 497, 786, 1207, 1942, 2905, 4498, 6703, 10574, 15597, 23754, 35043, 52422, 78369, 115522, 169499, 248150, 360521, 532466, 768275, 1116126, 1606669, 2314426, 3301879, 4777078, 6772657, 9677138, 13688079, 19406214

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

Also the number of binary words of length n starting with 1 and having all distinct runs (ranked by A175413, counted by A351016).

			The a(1) = 1 through a(6) = 18 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,1,2)  (4,1)    (4,2)
                     (2,1,1)  (1,1,3)  (5,1)
                              (1,2,2)  (1,1,4)
                              (2,2,1)  (1,2,3)
                              (3,1,1)  (1,3,2)
                                       (2,1,3)
                                       (2,3,1)
                                       (3,1,2)
                                       (3,2,1)
                                       (4,1,1)
                                       (1,1,2,2)
                                       (1,2,2,1)
                                       (2,1,1,2)
                                       (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The case of partitions is A000726.
The version for run-lengths instead of runs is A032020.
These words are ranked by A175413.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A116608 counts compositions by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329738 counts compositions with equal run-lengths.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A025047, A098504, A098859, A106356, A212322, A328592, A329740, A334028, A349054, A350952, A351205.

Table[Length[Select[Tuples[{0,1},n],#=={}||First[#]==1&&UnsameQ@@Split[#]&]],{n,0,10}]

P(n)=prod(k=1, n, 1 + y*x^k + O(x*x^n));
seq(n)=my(p=P(n)); Vec(sum(k=0, n, polcoef(p,k\2,y)*(k\2)!*polcoef(p,(k+1)\2,y)*((k+1)\2)!)) \\ Andrew Howroyd, Feb 11 2022

a(n>0) = A351016(n)/2.

G.f.: Sum_{k>=0} floor(k/2)! * ceiling(k/2)! * ([y^floor(k/2)] P(x,y)) * ([y^ceiling(k/2)] P(x,y)), where P(x,y) = Product_{k>=1} 1 + y*x^k. - Andrew Howroyd, Feb 11 2022

Terms a(21) and beyond from Andrew Howroyd, Feb 11 2022

A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

Original entry on oeis.org

13, 22, 25, 45, 46, 49, 53, 54, 59, 76, 77, 82, 89, 91, 93, 94, 97, 101, 102, 105, 108, 109, 110, 115, 118, 141, 148, 150, 153, 156, 162, 165, 166, 173, 177, 178, 180, 181, 182, 183, 187, 189, 190, 193, 197, 198, 201, 204, 205, 209, 210, 213, 214, 216, 217

Offset: 1

Gus Wiseman, Feb 12 2022

nonn

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
  13:     1101  (1,2,1)
  22:    10110  (2,1,2)
  25:    11001  (1,3,1)
  45:   101101  (2,1,2,1)
  46:   101110  (2,1,1,2)
  49:   110001  (1,4,1)
  53:   110101  (1,2,2,1)
  54:   110110  (1,2,1,2)
  59:   111011  (1,1,2,1,1)
  76:  1001100  (3,1,3)
  77:  1001101  (3,1,2,1)
  82:  1010010  (2,3,2)
  89:  1011001  (2,1,3,1)
  91:  1011011  (2,1,2,1,1)
  93:  1011101  (2,1,1,2,1)
  94:  1011110  (2,1,1,1,2)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for Heinz numbers of partitions is A130092, complement A130091.
Normal multisets with a permutation of this type appear to be A283353.
Partitions w/o permutations of this type are A351204, complement A351203.
The version using binary expansions is A351205, complement A175413.
The complement is A351290, counted by A351013.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has all distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612, counted by A003242.
A345167 ranks alternating compositions, counted by A025047.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions (A066099, reverse A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!UnsameQ@@Split[stc[#]]&]

A200649 Number of 1's in the Stolarsky representation of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 4, 1, 3, 3, 3, 5, 2, 2, 4, 2, 4, 4, 4, 6, 1, 3, 3, 3, 5, 3, 3, 5, 3, 5, 5, 5, 7, 2, 2, 4, 2, 4, 4, 4, 6, 2, 4, 4, 4, 6, 4, 4, 6, 4, 6, 6, 6, 8, 1, 3, 3, 3, 5, 3, 3, 5, 3, 5, 5, 5, 7, 3, 3, 5, 3, 5, 5, 5, 7, 3, 5, 5, 5, 7, 5, 5, 7, 5, 7, 7

Offset: 1

Casey Mongoven, Nov 19 2011

For the Stolarsky representation of n, see the C. Mongoven link.

Conjecture: a(n) is the length of row n-1 of A385886. To obtain it, first take maximal anti-run lengths of binary indices of each nonnegative integer (giving A384877), then remove all duplicate rows (giving A385886), and finally take the length of each remaining row. For sum instead of length we appear to have A200648. For runs minus 1 instead of anti-runs see A200650. - Gus Wiseman, Jul 21 2025

			The Stolarsky representation of 19 is 11101. This has 4 1's. So a(19) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Casey Mongoven, Description of Stolarsky Representations.

Cf. A072649, A130312, A135818, A200651.
For length instead of number of 1's we have A200648.
For 0's instead of 1's we have A200650.
Stolarsky representation is listed by A385888, ranks A200714.
A000120 counts 1's in binary expansion.
A384877 lists anti-run lengths of binary indices, duplicates removed A385886.
A384890 counts maximal anti-runs of binary indices, ranked by A385816.
Cf. A023758, A044813, A048793, A069010, A164707, A245562, A245563, A328592, A384893.

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
a[n_] := Count[stol[n], 1]; Array[a, 100] (* Amiram Eldar, Jul 07 2023 *)

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
a(n) = vecsum(stol(n)); \\ Amiram Eldar, Jul 07 2023

a(n) = a(n - A130312(n-1)) + (A072649(n-1) - A072649(n - A130312(n-1) - 1)) mod 2 for n > 2 with a(1) = 0, a(2) = 1. - Mikhail Kurkov, Oct 19 2021 [verification needed]

a(n) = A200648(n) - A200650(n). - Amiram Eldar, Jul 07 2023

More terms from Amiram Eldar, Jul 07 2023

A384879 Numbers whose binary indices have all distinct lengths of maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 25, 26, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 49, 50, 52, 53, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 88, 97, 98, 100, 101, 104, 105, 106, 128, 129, 130

Offset: 1

Gus Wiseman, Jun 17 2025

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 813 are {1,3,4,6,9,10}, with maximal anti-runs ((1,3),(4,6,9),(10)), with lengths (2,3,1), so 813 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    5:     101 ~ {1,3}
    8:    1000 ~ {4}
    9:    1001 ~ {1,4}
   10:    1010 ~ {2,4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   17:   10001 ~ {1,5}
   18:   10010 ~ {2,5}
   19:   10011 ~ {1,2,5}
   20:   10100 ~ {3,5}
   21:   10101 ~ {1,3,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}

Subsets of this type are counted by A384177, for runs A384175 (complement A384176).
These are the indices of strict rows in A384877, see A384878, A245563, A245562, A246029.
A000120 counts binary indices.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384890 counts maximal anti-runs in binary indices, runs A069010.
Cf. A023758, A044813, A048793, A164707, A242882, A243815, A325325, A328592, A384879, A384884, A384886, A384893.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],UnsameQ@@Length/@Split[bpe[#],#2!=#1+1&]&]

A200650 Number of 0's in Stolarsky representation of n.

Original entry on oeis.org

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

Offset: 1

Casey Mongoven, Nov 19 2011

For the Stolarsky representation of n, see the C. Mongoven link.
a(n+1), n >= 1, gives the size of the n-th generation of each of the "[male-female] pair of Fibonacci rabbits" in the Fibonacci rabbits tree read right-to-left by row, the first pair (the root) being the 0th generation. (Cf. OEIS Wiki link below.) - Daniel Forgues, May 07 2015
From Daniel Forgues, May 07 2015: (Start)
Concatenation of:
  0:                      1,
  1:                      0,
  2:                      0,
  3:                   1, 0,
  4:                1, 1, 0,
  5:          2, 1, 1, 1, 0,
  6: 2, 2, 1, 2, 1, 1, 1, 0,
(...),
where row n, n >= 3, is row n-1 prepended by incremented row n-2. (End)
For n >= 3, this algorithm yields the next F_n terms of the sequence, where F_n is the n-th Fibonacci number (A000045). Since it is asymptotic to (phi^n)/sqrt(5), the number of terms thus obtained grows exponentially at each step! - Daniel Forgues, May 22 2015
Conjecture: a(n) is one less than the length of row n-1 of A385817. To obtain it, first take maximal run lengths of binary indices of each nonnegative integer (giving A245563), then remove all duplicate rows (giving A385817), and finally take the length of each remaining row and subtract 1. For sum instead of length we appear to have A200648. For anti-runs instead of runs we appear to have A341259 = A200649-1. - Gus Wiseman, Jul 21 2025
How is this related to A117479? - R. J. Mathar, Aug 10 2025

			The Stolarsky representation of 19 is 11101. This has one 0. So a(19) = 1.

Kenny Lau, Table of n, a(n) for n = 1..20000
Casey Mongoven, Description of Stolarsky Representations.
OEIS Wiki, Fibonacci rabbits per generation.

For length instead of number of 0's we have A200648.
For sum (or number of 1's) instead of number of 0's we have A200649.
Stolarsky representation is listed by A385888, ranks A200714.
A000120 counts 1's in binary expansion, 0's A023416.
A069010 counts maximal runs of binary indices, ranked by A385889.
A245563 lists maximal run lengths of binary indices, duplicates removed A385817.
Cf. A000045, A023758, A135818, A200651, A245562, A328592, A385886.

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
a[n_] := If[n == 1, 1, Count[stol[n], 0]]; Array[a, 100] (* Amiram Eldar, Jul 07 2023 *)

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
a(n) = if(n == 1, 1,  my(s = stol(n)); #s - vecsum(s)); \\ Amiram Eldar, Jul 07 2023

a(n) = A200648(n) - A200649(n). - Amiram Eldar, Jul 07 2023

Corrected and extended by Kenny Lau, Jul 04 2016

A350952 The smallest number whose binary expansion has exactly n distinct runs.

Original entry on oeis.org

0, 1, 2, 11, 38, 311, 2254, 36079, 549790, 17593311, 549687102, 35179974591, 2225029922430, 284803830071167, 36240869367020798, 9277662557957324543, 2368116566113212692990, 1212475681849964898811391, 619877748107024946567312382, 634754814061593545284927880191

Offset: 0

Gus Wiseman, Feb 14 2022

nonn

Positions of first appearances in A297770 (with offset 0).

The binary expansion of terms for n > 0 starts with 1, then floor(n/2) 0's, then alternates runs of increasing numbers of 1's, and decreasing numbers of 0's; see Python code. Thus, for n even, terms have n*(n/2+1)/2 binary digits, and for n odd, ((n+1) + (n-1)*((n-1)/2+1))/2 binary digits. - Michael S. Branicky, Feb 14 2022

			The terms and their binary expansions begin:
       0:                   ()
       1:                    1
       2:                   10
      11:                 1011
      38:               100110
     311:            100110111
    2254:         100011001110
   36079:     1000110011101111
  549790: 10000110001110011110
For example, 311 has binary expansion 100110111 with 5 distinct runs: 1, 00, 11, 0, 111.

Michael S. Branicky, Table of n, a(n) for n = 0..114
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Runs in binary expansion are counted by A005811, distinct A297770.
The version for run-lengths instead of runs is A165933, for A165413.
Subset of A175413 (binary expansion has distinct runs), for lengths A044813.
The version for standard compositions is A351015.
A000120 counts binary weight.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A334028 counts distinct parts in standard compositions.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A070939, A085207, A098859, A233564, A325545, A328592, A333489, A351203, A351291.

q=Table[Length[Union[Split[If[n==0,{},IntegerDigits[n,2]]]]],{n,0,1000}];Table[Position[q,i][[1,1]]-1,{i,Union[q]}]

a(n)={my(t=0); for(k=1, (n+1)\2, t=((t<Andrew Howroyd, Feb 15 2022

def a(n): # returns term by construction
    if n == 0: return 0
    q, r = divmod(n, 2)
    if r == 0:
        s = "".join("1"*i + "0"*(q-i+1) for i in range(1, q+1))
        assert len(s) == n*(n//2+1)//2
    else:
        s = "1" + "".join("0"*(q-i+2) + "1"*i for i in range(2, q+2))
        assert len(s) == ((n+1) + (n-1)*((n-1)//2+1))//2
    return int(s, 2)
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 14 2022

a(9)-a(19) from Michael S. Branicky, Feb 14 2022

cp's OEIS Frontend

A351200 Number of patterns of length n with all distinct runs.

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A351290 Numbers k such that the k-th composition in standard order has all distinct runs.

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A351292 Number of patterns of length n with all distinct run-lengths.

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A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

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A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

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A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

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A200649 Number of 1's in the Stolarsky representation of n.

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