A274174 -id:A274174 - cp's OEIS frontend

A261983 Number of compositions of n such that at least two adjacent parts are equal.

0, 0, 1, 1, 4, 9, 18, 41, 89, 185, 388, 810, 1670, 3435, 7040, 14360, 29226, 59347, 120229, 243166, 491086, 990446, 1995410, 4016259, 8076960, 16231746, 32599774, 65437945, 131293192, 263316897, 527912140, 1058061751, 2120039885, 4246934012, 8505864640

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

			a(5) = 9: 311, 113, 221, 122, 2111, 1211, 1121, 1112, 11111.
From _Gus Wiseman_, Jul 07 2020: (Start)
The a(2) = 1 through a(6) = 18 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,2,2,1)
                             (1,1,1,1,1)  (1,3,1,1)
                                          (2,1,1,2)
                                          (2,2,1,1)
                                          (3,1,1,1)
                                          (1,1,1,1,2)
                                          (1,1,1,2,1)
                                          (1,1,2,1,1)
                                          (1,2,1,1,1)
                                          (2,1,1,1,1)
                                          (1,1,1,1,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=1 of A261981.
Cf. A011782, A262046.
The complement A003242 counts anti-runs.
Sum of positive-indexed terms of row n of A106356.
Row sums of A131044.
The (1,1,1) matching case is A335464.
Strict compositions are A032020.
Compositions with adjacent parts coprime are A167606.
Compositions with equal parts contiguous are A274174.
Cf. A114901, A178470, A242882, A244164, A325534, A335448, A335452.

b:= proc(n, i) option remember; `if`(n=0, 0, add(
      `if`(i=j, ceil(2^(n-j-1)), b(n-j, j)), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}] (* Gus Wiseman, Jul 06 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 0, Sum[If[i == j, Ceiling[2^(n-j-1)], b[n-j, j]], {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz's Maple code *)

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 08 2015

a(n) = A011782(n) - A003242(n). - Emeric Deutsch, Jul 03 2020

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 143, 168, 170, 175, 187, 238, 248, 250, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047, 2048, 2080, 2084, 2090, 2111, 2184

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into non-overlapping runs, read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

The k-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 k, 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)
     7:     111  (1,1,1)
     8:    1000  (4)
    10:    1010  (2,2)
    11:    1011  (2,1,1)
    14:    1110  (1,1,2)
    15:    1111  (1,1,1,1)
    16:   10000  (5)
    31:   11111  (1,1,1,1,1)
    32:  100000  (6)
    36:  100100  (3,3)
    39:  100111  (3,1,1,1)
    42:  101010  (2,2,2)
    46:  101110  (2,1,1,2)
    59:  111011  (1,1,2,1,1)
    60:  111100  (1,1,1,3)
For example:
- The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
- The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
- The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.

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

Standard compositions are listed by A066099.
For equal lengths instead of sums we have A353744, counted by A329738.
The version for partitions is A353833, counted by A304442.
These compositions are counted by A353851.
The distinct instead of equal version is A353852, counted by A353850.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.

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

A353849(a(n)) = 1.

A374632 Number of integer compositions of n whose leaders of weakly increasing runs are distinct.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 40, 69, 119, 200, 335, 557, 917, 1499, 2433, 3920, 6280, 10004, 15837, 24946, 39087, 60952, 94606, 146203, 224957, 344748, 526239, 800251, 1212527, 1830820, 2754993, 4132192, 6178290, 9209308, 13686754, 20282733, 29973869, 44175908, 64936361

Offset: 0

Gus Wiseman, Jul 23 2024

nonn

The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

			The composition (4,2,2,1,1,3) has weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so is counted under a(13).
The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (211)   (113)
                        (1111)  (122)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by A374768 = positions of distinct rows in A374629 (sums A374630).
Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A274174, ranks A374249.
- For leaders of anti-runs we have A374518, ranks A374638.
- For leaders of strictly increasing runs we have A374687, ranks A374698.
- For leaders of weakly decreasing runs we have A374743, ranks A335467.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Types of run-leaders (instead of distinct):
- For strictly decreasing leaders we appear to have A188920.
- For weakly decreasing leaders we appear to have A189076.
- For identical leaders we have A374631.
- For weakly increasing leaders we have A374635.
- For strictly increasing leaders we have A374634.
A003242 counts anti-run compositions.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Cf. A106356, A124766, A238343, A261982, A333213, A335548, A373949, A373953.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,LessEqual]&]],{n,0,15}]

dfs(m, r, v) = 1 + sum(s=1, min(m, r-1), if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s/(1-x^s) + sum(t=s+1, m-s, dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)/prod(i=s, t, 1-x^i))));
lista(nn) = Vec(dfs(nn, nn+1, []) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A374635 Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 36, 69, 130, 247, 467, 890, 1689, 3213, 6110, 11627, 22121, 42101, 80124, 152512, 290300, 552609, 1051953, 2002583, 3812326, 7257679, 13816867, 26304254, 50077792, 95338234, 181505938, 345554234, 657874081, 1252478707, 2384507463, 4539705261

Offset: 0

Gus Wiseman, Jul 23 2024

nonn

The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

			The composition (1,3,3,2,4,2) has weakly increasing runs ((1,3,3),(2,4),(2)), with leaders (1,2,2), so is counted under a(15).
The a(0) = 1 through a(6) = 20 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (111)  (22)    (23)     (24)
                        (112)   (113)    (33)
                        (121)   (122)    (114)
                        (1111)  (131)    (123)
                                (1112)   (132)
                                (1121)   (141)
                                (1211)   (222)
                                (11111)  (1113)
                                         (1122)
                                         (1131)
                                         (1212)
                                         (1221)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by positions of weakly increasing rows in A374629 (sums A374630).
Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A000041.
- For leaders of weakly decreasing runs we have A188900.
- For leaders of anti-runs we have A374681.
- For leaders of strictly increasing runs we have A374690.
- For leaders of strictly decreasing runs we have A374764.
Types of run-leaders (instead of weakly increasing):
- For strictly decreasing leaders we appear to have A188920.
- For weakly decreasing leaders we appear to have A189076.
- For identical leaders we have A374631.
- For distinct leaders we have A374632, ranks A374768.
- For strictly increasing leaders we have A374634.
A003242 counts anti-run compositions.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Cf. A106356, A124766, A238343, A261982, A333213, A374518, A374687, A374761.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,LessEqual]&]],{n,0,15}]

dfs(m, r, u) = 1 + sum(s=u, min(m, r-1), x^s/(1-x^s) + sum(t=s+1, m-s, dfs(m-s-t, t, s)*x^(s+t)/prod(i=s, t, 1-x^i)));
lista(nn) = Vec(dfs(nn, nn+1, 1) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 10 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			Triangle begins:
  1
  2
  2
  3
  2 1
  1 2
  3
  4
  3 1
  4
  2 2
  1 3
  1 2 1
For example, composition 350 in standard order is (2,2,1,1,1,2), so row 350 is (4,3,2).

Row-sums are A029837.
Standard compositions are listed by A066099.
Row-lengths are A124767.
These compositions are ranked by A353847.
Row k has A353849(k) distinct parts.
The version for partitions is A354584, ranked by A353832.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353863, A353864.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total/@Split[stc[n]],{n,0,30}]

A374683 Irregular triangle read by rows where row n lists the leaders of strictly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 26 2024

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

			The maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so row 1234567 is (3,2,1,2,1,1,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly increasing runs begin:
   0:      () -> ()         15: (1,1,1,1) -> (1,1,1,1)
   1:     (1) -> (1)        16:       (5) -> (5)
   2:     (2) -> (2)        17:     (4,1) -> (4,1)
   3:   (1,1) -> (1,1)      18:     (3,2) -> (3,2)
   4:     (3) -> (3)        19:   (3,1,1) -> (3,1,1)
   5:   (2,1) -> (2,1)      20:     (2,3) -> (2)
   6:   (1,2) -> (1)        21:   (2,2,1) -> (2,2,1)
   7: (1,1,1) -> (1,1,1)    22:   (2,1,2) -> (2,1)
   8:     (4) -> (4)        23: (2,1,1,1) -> (2,1,1,1)
   9:   (3,1) -> (3,1)      24:     (1,4) -> (1)
  10:   (2,2) -> (2,2)      25:   (1,3,1) -> (1,1)
  11: (2,1,1) -> (2,1,1)    26:   (1,2,2) -> (1,2)
  12:   (1,3) -> (1)        27: (1,2,1,1) -> (1,1,1)
  13: (1,2,1) -> (1,1)      28:   (1,1,3) -> (1,1)
  14: (1,1,2) -> (1,1)      29: (1,1,2,1) -> (1,1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Row-leaders are A065120.
Row-lengths are A124768.
Other types of runs: A374251, A374515, A374740.
The weak version is A374629, sum A374630, length A124766.
Row-sums are A374684.
Positions of identical rows are A374685, counted by A374686.
Positions of distinct (strict) rows are A374698, counted by A374687.
The opposite version is A374757, sum A374758, length A124769.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124767, A333381.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374700.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[First/@Split[stc[n],Less],{n,0,100}]

A374740 Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 24 2024

The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

			The maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so row 1234567 is (3,2,2,5).
The nonnegative integers, corresponding compositions, and leaders of weakly decreasing runs begin:
    0: () -> ()           15: (1,1,1,1) -> (1)
    1: (1) -> (1)         16: (5) -> (5)
    2: (2) -> (2)         17: (4,1) -> (4)
    3: (1,1) -> (1)       18: (3,2) -> (3)
    4: (3) -> (3)         19: (3,1,1) -> (3)
    5: (2,1) -> (2)       20: (2,3) -> (2,3)
    6: (1,2) -> (1,2)     21: (2,2,1) -> (2)
    7: (1,1,1) -> (1)     22: (2,1,2) -> (2,2)
    8: (4) -> (4)         23: (2,1,1,1) -> (2)
    9: (3,1) -> (3)       24: (1,4) -> (1,4)
   10: (2,2) -> (2)       25: (1,3,1) -> (1,3)
   11: (2,1,1) -> (2)     26: (1,2,2) -> (1,2)
   12: (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2)
   13: (1,2,1) -> (1,2)   28: (1,1,3) -> (1,3)
   14: (1,1,2) -> (1,2)   29: (1,1,2,1) -> (1,2)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Row-leaders are A065120.
Row-lengths are A124765.
Other types of runs are A374251, A374515, A374683, A374757.
The opposite is A374629.
Positions of distinct (strict) rows are A374701, counted by A374743.
Row-sums are A374741, opposite A374630.
Positions of identical rows are A374744, counted by A374742.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[First/@Split[stc[n],GreaterEqual],{n,0,100}]

A374515 Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 31 2024

Anti-runs summing to n are counted by A003242(n).
The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
The k-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 k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The maximal anti-runs of the 1234567th composition in standard order are ((3,2,1,2),(2,1,2,5,1),(1),(1)), so row 1234567 is (3,2,1,1).
The nonnegative integers, corresponding compositions, and leaders of anti-runs begin:
    0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
    1:     (1) -> (1)       16:       (5) -> (5)
    2:     (2) -> (2)       17:     (4,1) -> (4)
    3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
    4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2)       20:     (2,3) -> (2)
    6:   (1,2) -> (1)       21:   (2,2,1) -> (2,2)
    7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2)
    8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
    9:   (3,1) -> (3)       24:     (1,4) -> (1)
   10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1)
   11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2)
   12:   (1,3) -> (1)       27: (1,2,1,1) -> (1,1)
   13: (1,2,1) -> (1)       28:   (1,1,3) -> (1,1)
   14: (1,1,2) -> (1,1)     29: (1,1,2,1) -> (1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Row-leaders of nonempty rows are A065120.
Row-lengths are A333381.
Row-sums are A374516.
Positions of identical rows are A374519 (counted by A374517).
Positions of distinct (strict) rows are A374638 (counted by A374518).
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression is A373948 or A374251, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A059893, A228351, A233564, A272919, A333213, A373949.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[First/@Split[stc[n],UnsameQ],{n,0,100}]

A374518 Number of integer compositions of n whose leaders of anti-runs are distinct.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 32, 58, 112, 201, 371, 694, 1276, 2342, 4330, 7958, 14613, 26866, 49303, 90369, 165646, 303342, 555056, 1015069, 1855230

Offset: 0

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(6) = 17 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                      (211)  (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                             (311)  (213)
                                    (231)
                                    (312)
                                    (321)
                                    (411)
                                    (1212)
                                    (1221)
                                    (2112)
                                    (2121)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

These compositions have ranks A374638.
The complement is counted by A374678.
For partitions instead of compositions we have A375133.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A274174, ranks A374249.
- For leaders of weakly increasing runs we have A374632, ranks A374768.
- For leaders of strictly increasing runs we have A374687, ranks A374698.
- For leaders of weakly decreasing runs we have A374743, ranks A374701.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Other types of run-leaders (instead of distinct):
- For identical leaders we have A374517.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
Cf. A188920, A233564, A238343, A333213, A333381, A373949, A374515.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]

A005314 For n = 0, 1, 2, a(n) = n; thereafter, a(n) = 2*a(n-1) - a(n-2) + a(n-3).

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 16, 28, 49, 86, 151, 265, 465, 816, 1432, 2513, 4410, 7739, 13581, 23833, 41824, 73396, 128801, 226030, 396655, 696081, 1221537, 2143648, 3761840, 6601569, 11584946, 20330163, 35676949, 62608681, 109870576, 192809420, 338356945, 593775046

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts congruent to {1,2} mod 4. - Vladeta Jovovic, Mar 10 2005
a(n)/a(n-1) tends to A109134; an eigenvalue of the matrix M and a root to the characteristic polynomial. - Gary W. Adamson, May 25 2007
Starting with offset 1 = INVERT transform of (1, 1, 0, 0, 1, 1, 0, 0, ...). - Gary W. Adamson, May 04 2009
a(n-2) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 0; 0, 1, 1; 1, 0, 1] or of the 3 X 3 matrix [0, 0, 1; 1, 1, 0; 0, 1, 1]. - R. J. Mathar, Feb 03 2014
Counts closed walks of length (n+2) at a vertex of a unidirectional triangle containing a loop on remaining two vertices. - David Neil McGrath, Sep 15 2014
Also the number of binary words of length n that begin with 1 and avoid the subword 101. a(5) = 9: 10000, 10001, 10010, 10011, 11000, 11001, 11100, 11110, 11111. - Alois P. Heinz, Jul 21 2016
Also the number of binary words of length n-1 such that every two consecutive 0s are immediately followed by at least two consecutive 1s. a(4) = 5: 010, 011, 101, 110, 111. - Jerrold Grossman, May 03 2024

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 16*x^6 + 28*x^7 + 49*x^8 + ...
From _Gus Wiseman_, Nov 25 2019: (Start)
a(n) is the number of subsets of {1..n} containing n such that if x and x + 2 are both in the subset, then so is x + 1. For example, the a(1) = 1 through a(5) = 9 subsets are:
  {1}  {2}    {3}      {4}        {5}
       {1,2}  {2,3}    {1,4}      {1,5}
              {1,2,3}  {3,4}      {2,5}
                       {2,3,4}    {4,5}
                       {1,2,3,4}  {1,2,5}
                                  {1,4,5}
                                  {3,4,5}
                                  {2,3,4,5}
                                  {1,2,3,4,5}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..400
Isha Agarwal, Matvey Borodin, Aidan Duncan, Kaylee Ji, Tanya Khovanova, Shane Lee, Boyan Litchev, Anshul Rastogi, Garima Rastogi, and Andrew Zhao, From Unequal Chance to a Coin Game Dance: Variants of Penney's Game, arXiv:2006.13002 [math.HO], 2020.
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
Hung Viet Chu and Zachary Louis Vasseur, Schreier Sets of Multiples of an Integer, Linear Recurrence, and Pascal Triangle, arXiv:2506.14312 [math.CO], 2025. See Table 1 p. 2.
Christian Ennis, William Holland, Omer Mujawar, Aadit Narayanan, Frank Neubrander, Marie Neubrander, and Christina Simino, Words in Random Binary Sequences I, arXiv:2107.01029 [math.GM], 2021.
R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.
Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See pp. 8, 10.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 426
L. A. Medina and A. Straub, On multiple and infinite log-concavity, 2013, preprint Annals of Combinatorics, March 2016, Volume 20, Issue 1, pp 125-138.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Bojan Vučković and Miodrag Živković, Row Space Cardinalities Above 2^(n - 2) + 2^(n - 3), ResearchGate, January 2017, p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Equals row sums of triangle A099557.
Equals row sums of triangle A224838.
Cf. A011973 (starting with offset 1 = Falling diagonal sums of triangle with rows displayed as centered text).
First differences of A005251, shifted twice to the left.
Cf. A003242, A114901, A261041, A274174.

a005314 n = a005314_list !! n
a005314_list = 0 : 1 : 2 : zipWith (+) a005314_list
   (tail $ zipWith (-) (map (2 *) $ tail a005314_list) a005314_list)
-- Reinhard Zumkeller, Oct 14 2011

[0] cat [n le 3 select n else 2*Self(n-1) - Self(n-2) + Self(n-3):n in [1..35]]; // Marius A. Burtea, Oct 24 2019

R:=PowerSeriesRing(Integers(), 36); [0] cat Coefficients(R!( x/(1-2*x+x^2-x^3))); // Marius A. Burtea, Oct 24 2019

A005314 := proc(n)
    option remember ;
    if n <=2 then
        n;
    else
        2*procname(n-1)-procname(n-2)+procname(n-3) ;
    end if;
end proc:
seq(A005314(n),n=0..20) ; # R. J. Mathar, Feb 25 2024

LinearRecurrence[{2, -1, 1}, {0, 1, 2}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
Table[Sum[Binomial[n - Floor[(k + 1)/2], n - Floor[(3 k - 1)/2]], {k, 0, n}], {n, 0, 100}] (* John Molokach, Jul 21 2013 *)
Table[Sum[Binomial[n - Floor[(4 n + 15 - 6 k + (-1)^k)/12], n - Floor[(4 n + 15 - 6 k + (-1)^k)/12] - Floor[(2 n - 1)/3] + k - 1], {k, 1, Floor[(2 n + 2)/3]}], {n, 0, 100}] (* John Molokach, Jul 25 2013 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ x^2 / (1 - x + 2 x^2 - x^3), {x, 0, -n}], SeriesCoefficient[ x / (1 - 2 x + x^2 - x^3), {x, 0, n}]]; (* Michael Somos, Dec 13 2013 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[n]==2a[n-1]-a[n-2]+a[n-3]},a,{n,40}] (* Harvey P. Dale, May 13 2018 *)
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!MatchQ[#,{_,x_,y_,_}/;x+2==y]&]],{n,0,10}] (* Gus Wiseman, Nov 25 2019 *)

{a(n) = sum(k=0, (2*n-1)\3, binomial(n-1-k\2, k))}

{a(n) = if( n<0, polcoeff( x^2 / (1 - x + 2*x^2 - x^3) + x * O(x^-n), -n), polcoeff( x / (1 - 2*x + x^2 - x^3) + x * O(x^n), n))}; /* Michael Somos, Sep 18 2012 */

def A005314(n): return sum( binomial(n-k, 2*k+1) for k in range(floor((n+2)/3)) )
[A005314(n) for n in range(51)] # G. C. Greubel, Nov 10 2023

From Paul D. Hanna, Oct 22 2004: (Start)
G.f.: x/(1-2*x+x^2-x^3).
a(n) = Sum_{k=0..[(2n-1)/3]} binomial(n-1-[k/2], k), where [x]=floor(x). (End)
a(n) = Sum_{k=0..n} binomial(n-k, 2*k+1).
23*a_n = 3*P_{2n+2} + 7*P_{2n+1} - 2*P_{2n}, where P_n are the Perrin numbers, A001608. - Don Knuth, Dec 09 2008
G.f. (1-z)*(1+z^2)/(1-2*z+z^2-z^3) for the augmented version 1, 1, 2, 3, 5, 9, 16, 28, 49, 86, 151, ... was given in Simon Plouffe's thesis of 1992.
a(n) = a(n-1) + a(n-2) + a(n-4) = a(n-2) + A049853(n-1) = a(n-1) + A005251(n) = Sum_{i <= n} A005251(i).
a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-k, 2*k+1). - Richard L. Ollerton, May 12 2004
M^n*[1,0,0] = [a(n-2), a(n-1), a]; where M = the 3 X 3 matrix [0,1,0; 0,0,1; 1,-1,2]. Example M^5*[1,0,0] = [3,5,9]. - Gary W. Adamson, May 25 2007
a(n) = A000931(2*n + 4). - Michael Somos, Sep 18 2012
a(n) = A077954(-n - 2). - Michael Somos, Sep 18 2012
G.f.: 1/( 1 - Sum_{k>=0} x*(x-x^2+x^3)^k ) - 1. - Joerg Arndt, Sep 30 2012
a(n) = Sum_{k=0..n} binomial( n-floor((k+1)/2), n-floor((3k-1)/2) ). - John Molokach, Jul 21 2013
a(n) = Sum_{k=1..floor((2*n+2)/3)} binomial(n - floor((4*n+15-6*k+(-1)^k)/12), n - floor((4*n+15-6*k+(-1)^k)/12) - floor((2*n-1)/3) + k - 1). - John Molokach, Jul 24 2013
a(n) = round(A001608(2n+1)*r) where r is the real root of 23*x^3 - 23*x^2 + 8*x - 1 = 0, r = 0.4114955... - Richard Turk, Oct 24 2019
a(n+2) = n + 2 + Sum_{k=0..n} (n-k)*a(k). - Greg Dresden and Yichen P. Wang, Sep 16 2021
a(n) ~ (19 - r + 11*r^2) / (23 * r^(n-1)), where r = 0.569840290998... is the root of the equation r*(2 - r + r^2) = 1. - Vaclav Kotesovec, Apr 14 2024
a(n) = n*3F2(1/3-n/3,2/3-n/3,1-n/3;-n,3/2;27/4). - R. J. Mathar, Jun 27 2024
If p,q,r are the three solutions to x^3 = 2x^2 - x + 1, then a(n) = p^(n+1)/((p-q)*(p-r)) + q^(n+1)/((q-p)*(q-r)) + r^(n+1)/((r-p)*(r-q)). Compare to similar formula for A005251. - Greg Dresden and AnXing Yang, Aug 19 2025

More terms and additional formulas from Henry Bottomley, Jul 21 2000

Plouffe's g.f. edited by R. J. Mathar, May 12 2008

cp's OEIS Frontend

A261983 Number of compositions of n such that at least two adjacent parts are equal.

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A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

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A374632 Number of integer compositions of n whose leaders of weakly increasing runs are distinct.

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A374635 Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.

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A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

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A374683 Irregular triangle read by rows where row n lists the leaders of strictly increasing runs in the n-th composition in standard order.

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A374740 Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.

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A374515 Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.

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