A333256 -id:A333256 - cp's OEIS frontend

A124763 Number of non-rises (levels or falls) for compositions in standard order.

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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. a(n) is one fewer than the number of maximal strictly increasing runs in this composition. Alternatively, a(n) is the number of weak descents in the same composition. For example, the strictly increasing runs of the 1234567th composition are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so a(1234567) = 8 - 1 = 7. The 7 weak descents together with the strict ascents are: 3 >= 2 >= 1 < 2 >= 2 >= 1 < 2 < 5 >= 1 >= 1 >= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; 2>=1>=1, so a(11) = 2.
The table starts:
  0
  0
  0 1
  0 1 0 2
  0 1 1 2 0 1 1 3
  0 1 1 2 0 2 1 3 0 1 1 2 1 2 2 4
  0 1 1 2 1 2 1 3 0 1 2 3 1 2 2 4 0 1 1 2 0 2 1 3 1 2 2 3 2 3 3 5

Cf. A029931, A066099, A124760, A124761, A124764, A011782 (row lengths), A045883 (row sums), A238343, A333220.
Compositions of n with k weak descents are A333213.
Positions of zeros are A333255.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],GreaterEqual@@#&]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1=b(i+1)} 1.
a(n) = A124761(n) + A124762(n).
For n > 0, a(n) = A124768(n) - 1. - Gus Wiseman, Apr 08 2020

A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

Original entry on oeis.org

0, 3, 10, 15, 36, 43, 58, 63, 136, 147, 170, 175, 228, 235, 250, 255, 528, 547, 586, 591, 676, 683, 698, 703, 904, 915, 938, 943, 996, 1003, 1018, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2362, 2367, 2696, 2707, 2730, 2735, 2788, 2795, 2810, 2815, 3600, 3619

Offset: 1

Gus Wiseman, Feb 01 2022

nonn

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 the corresponding compositions begin:
    0:         0  ()
    3:        11  (1,1)
   10:      1010  (2,2)
   15:      1111  (1,1,1,1)
   36:    100100  (3,3)
   43:    101011  (2,2,1,1)
   58:    111010  (1,1,2,2)
   63:    111111  (1,1,1,1,1,1)
  136:  10001000  (4,4)
  147:  10010011  (3,3,1,1)
  170:  10101010  (2,2,2,2)
  175:  10101111  (2,2,1,1,1,1)
  228:  11100100  (1,1,3,3)
  235:  11101011  (1,1,2,2,1,1)
  250:  11111010  (1,1,1,1,2,2)
  255:  11111111  (1,1,1,1,1,1,1,1)

The case of twins (binary weight 2) is A000120.
The Heinz numbers of these compositions are given by A000290.
All terms are evil numbers A001969.
Partitions of this type are counted by A035363, any length A351004.
These compositions are counted by A077957(n-2), see also A016116.
The strict case (distinct twins) is A351009, counted by A032020 with 0's.
The anti-run case is A351011, counted by A003242 interspersed with 0's.
A011782 counts integer compositions.
A085207/A085208 represent concatenation of standard compositions.
A333489 ranks anti-runs, complement A348612.
A345167/A350355/A350356 rank alternating compositions.
A351014 counts distinct runs in standard compositions.
Cf. A018819, A025047, A027383, A035457, A053738, A088218, A106356, A238279, A344604, A351012, A351015.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- 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.

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

A156242 Bisection of A054353.

Original entry on oeis.org

3, 6, 9, 12, 15, 19, 21, 24, 27, 30, 33, 36, 39, 42, 45, 47, 50, 54, 57, 60, 63, 66, 69, 72, 75, 77, 81, 84, 87, 90, 93, 96, 100, 102, 105, 108, 111, 114, 117, 120, 123, 127, 129, 132, 136, 139, 142, 145, 147, 151, 154, 156, 159, 163, 166, 169, 172, 174, 177, 181

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

Positions of strict descents in the Kolakoski sequence A000002. Strict ascents are A156243. - Gus Wiseman, Mar 31 2020

Cf. A000002, A156243.
The version for prime gaps is A258026.
Sizes of maximal weakly increasing subsequences of A000002 are A332875.
Cf. A054804, A124766, A124769, A225620, A238343, A332273, A333213, A333215, A333256, A333383.

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=Nest[kolagrow,{1},n-1];
Join@@Position[Partition[kol[100],2,1],{2,1}] (* Gus Wiseman, Mar 31 2020 *)

a(n) = A054353(2n).

A000002(a(n))=2 and A000002(a(n)+1)=1. - Jon Perry, Sep 04 2012

A350355 Numbers k such that the k-th composition in standard order is up/down.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 13, 16, 20, 24, 25, 32, 40, 41, 48, 49, 50, 54, 64, 72, 80, 81, 82, 96, 97, 98, 102, 108, 109, 128, 144, 145, 160, 161, 162, 166, 192, 193, 194, 196, 198, 204, 205, 216, 217, 256, 272, 288, 289, 290, 320, 321, 322, 324, 326, 332, 333, 384

Offset: 1

Gus Wiseman, Jan 15 2022

nonn

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.

A composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2).

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   6: (1,2)
   8: (4)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  32: (6)
  40: (2,4)
  41: (2,3,1)
  48: (1,5)
  49: (1,4,1)
  50: (1,3,2)
  54: (1,2,1,2)

The case of permutations is counted by A000111.
These compositions are counted by A025048, down/up A025049.
The strict case is counted by A129838, undirected A349054.
The weak version is counted by A129852, down/up A129853.
The version for anti-runs is A333489, a superset, complement A348612.
This is the up/down case of A345167, counted by A025047.
Counting patterns of this type gives A350354.
The down/up version is A350356.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions, unordered A000041.
A345192 counts non-alternating compositions, ranked by A345168.
A349052 counts weakly alternating compositions, complement A349053.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.

updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]

A345167 = A350355 \/ A350356.

A350356 Numbers k such that the k-th composition in standard order is down/up.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 22, 32, 33, 34, 38, 44, 45, 64, 65, 66, 68, 70, 76, 77, 88, 89, 128, 129, 130, 132, 134, 140, 141, 148, 152, 153, 176, 177, 178, 182, 256, 257, 258, 260, 262, 264, 268, 269, 276, 280, 281, 296, 297, 304, 305, 306, 310, 352, 353

Offset: 1

Gus Wiseman, Jan 15 2022

nonn

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.

A composition is down/up if it is alternately strictly increasing and strictly decreasing, starting with a decrease. For example, the partition (3,2,2,2,1) has no down/up permutations, even though it does have the anti-run permutation (2,1,2,3,2).

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   8: (4)
   9: (3,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  22: (2,1,2)
  32: (6)
  33: (5,1)
  34: (4,2)
  38: (3,1,2)
  44: (2,1,3)
  45: (2,1,2,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions
Wikipedia, Alternating permutation

The case of permutations is counted by A000111.
These compositions are counted by A025049, up/down A025048.
The strict case is counted by A129838, undirected A349054.
The weak version is counted by A129853, up/down A129852.
The version for anti-runs is A333489, a superset, complement A348612.
This is the down/up case of A345167, counted by A025047.
Counting patterns of this type gives A350354.
The up/down version is A350355.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions, unordered A000041.
A345192 counts non-alternating compositions, ranked by A345168.
A349052 counts weakly alternating compositions, complement A349053.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.

doupQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]y[[m+1]]],{m,1,Length[y]-1}];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],doupQ[stc[#]]&]

A345167 = A350355 \/ A350356.

A337481 Number of compositions of n that are neither strictly increasing nor strictly decreasing.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 25, 55, 117, 241, 493, 1001, 2019, 4061, 8149, 16331, 32705, 65461, 130981, 262037, 524161, 1048425, 2096975, 4194097, 8388365, 16776933, 33554103, 67108481, 134217285, 268434945, 536870321, 1073741145, 2147482869, 4294966401, 8589933569

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

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

			The a(2) = 1 through a(5) = 11 compositions:
  (11)  (111)  (22)    (113)
               (112)   (122)
               (121)   (131)
               (211)   (212)
               (1111)  (221)
                       (311)
                       (1112)
                       (1121)
                       (1211)
                       (2111)
                       (11111)

Ranked by the complement of the intersection of A333255 and A333256.
A332834 is the weak version.
A337482 is the semi-strict version.
A337484 counts only compositions of length 3.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A218004 counts strictly increasing or weakly decreasing compositions.
Cf. A216652, A329398, A337462, A337483, A337605.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!Greater@@#&]],{n,0,15}]

a(n) = 2^(n-1) - 2*A000009(n) + 1, n > 0.

A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

Original entry on oeis.org

0, 9, 130, 135, 141, 153, 177, 193, 225, 2052, 2059, 2062, 2069, 2074, 2079, 2089, 2098, 2103, 2109, 2129, 2146, 2151, 2157, 2169, 2209, 2242, 2247, 2253, 2265, 2289, 2369, 2434, 2439, 2445, 2457, 2481, 2529, 2561, 2689, 2818, 2823, 2829, 2841, 2865, 2913

Offset: 1

Gus Wiseman, Nov 22 2021

nonn

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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The terms and corresponding compositions begin:
     0: ()
     9: (3,1)
   130: (6,2)
   135: (5,1,1,1)
   141: (4,1,2,1)
   153: (3,1,3,1)
   177: (2,1,4,1)
   193: (1,6,1)
   225: (1,1,5,1)
  2052: (9,3)
  2059: (8,2,1,1)
  2062: (8,1,1,2)
  2069: (7,2,2,1)
  2074: (7,1,2,2)
  2079: (7,1,1,1,1,1)
  2089: (6,2,3,1)
  2098: (6,1,3,2)
  2103: (6,1,2,1,1,1)

These compositions are counted by A224274 up to 0's.
An unordered version is A348617, counted by A001523 up to 0's.
The positive version is A349153, unreversed A348614.
The unreversed version is A349154.
Positive unordered unreversed: A349159, counted by A000712 up to 0's.
A positive unordered version is A349160, counted by A006330 up to 0's.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- Heinz number is given by A333219.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Alternating compositions are ranked by A345167, complement A345168.

stc[n_]:=Differences[Prepend[ Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,1000],Total[stc[#]]==-2*sats[stc[#]]&]

A351011 Numbers k such that the k-th composition in standard order has even length and alternately equal and unequal parts, i.e., all run-lengths equal to 2.

Original entry on oeis.org

0, 3, 10, 36, 43, 58, 136, 147, 228, 235, 528, 547, 586, 676, 698, 904, 915, 2080, 2115, 2186, 2347, 2362, 2696, 2707, 2788, 2795, 3600, 3619, 3658, 3748, 3770, 8256, 8323, 8458, 8740, 8747, 8762, 9352, 9444, 9451, 10768, 10787, 10826, 11144, 11155, 14368

Offset: 1

Gus Wiseman, Feb 03 2022

nonn

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 standard compositions begin:
    0:           0  ()
    3:          11  (1,1)
   10:        1010  (2,2)
   36:      100100  (3,3)
   43:      101011  (2,2,1,1)
   58:      111010  (1,1,2,2)
  136:    10001000  (4,4)
  147:    10010011  (3,3,1,1)
  228:    11100100  (1,1,3,3)
  235:    11101011  (1,1,2,2,1,1)
  528:  1000010000  (5,5)
  547:  1000100011  (4,4,1,1)
  586:  1001001010  (3,3,2,2)
  676:  1010100100  (2,2,3,3)
  698:  1010111010  (2,2,1,1,2,2)
  904:  1110001000  (1,1,4,4)
  915:  1110010011  (1,1,3,3,1,1)

The case of twins (binary weight 2) is A000120.
All terms are evil numbers A001969.
These compositions are counted by A003242 interspersed with 0's.
Partitions of this type are counted by A035457, any length A351005.
The Heinz numbers of these compositions are A062503.
Taking singles instead of twins gives A333489, complement A348612.
This is the anti-run case of A351010.
The strict case (distinct twins) is A351009, counted by A077957(n-2).
A011782 counts compositions.
A085207/A085208 represent concatenation of standard compositions.
A345167 ranks alternating compositions, counted by A025047.
A350355 ranks up/down compositions, counted by A025048.
A350356 ranks down/up compositions, counted by A025049.
A351014 counts distinct runs in standard compositions.
Cf. A008965, A018819, A027383, A032020, A035363, A088218, A106356, A122129, A122134, A238279, A351007.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- 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.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],And@@(#==2&)/@Length/@Split[stc[#]]&]

A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

Original entry on oeis.org

0, 12, 160, 193, 195, 198, 204, 216, 240, 2304, 2561, 2563, 2566, 2572, 2584, 2608, 2656, 2752, 2944, 3074, 3077, 3079, 3082, 3085, 3087, 3092, 3097, 3099, 3102, 3112, 3121, 3123, 3126, 3132, 3152, 3169, 3171, 3174, 3180, 3192, 3232, 3265, 3267, 3270, 3276

Offset: 1

Gus Wiseman, Nov 21 2021

nonn

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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The terms and corresponding compositions begin:
       0: ()
      12: (1,3)
     160: (2,6)
     193: (1,6,1)
     195: (1,5,1,1)
     198: (1,4,1,2)
     204: (1,3,1,3)
     216: (1,2,1,4)
     240: (1,1,1,5)
    2304: (3,9)
    2561: (2,9,1)
    2563: (2,8,1,1)
    2566: (2,7,1,2)
    2572: (2,6,1,3)
    2584: (2,5,1,4)

These compositions are counted by A224274 up to 0's.
Except for 0, a subset of A345919.
The positive version is A348614, reverse A349153.
An unordered version is A348617, counted by A001523.
The reverse version is A349155.
A positive unordered version is A349159, counted by A000712 up to 0's.
A000346 = even-length compositions with alt sum != 0, complement A001700.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Necklaces are ranked by A065609, dual A333764, reversed A333943.
- Alternating compositions are ranked by A345167, complement A345168.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Total[stc[#]]==-2*ats[stc[#]]&]

A333767 Length of shortest run of zeros after a one in the binary expansion of n. a(0) = 0.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 06 2020

nonn

			The binary expansion of 148 is (1,0,0,1,0,1,0,0), so a(148) = 1.

Positions of first appearances (ignoring index 0) are A000079.
Positions of terms > 0 are A022340.
Minimum prime index is A055396.
The  maximum part minus 1 is given by A087117.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without 1's are A022340.
- Sum is A070939.
- Product is A124758.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Maximum is A333766.
- Minimum is A333768.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A228351, A328594, A333217, A333218, A333219, A333632.

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

For n > 0, a(n) = A333768(n) - 1.

cp's OEIS Frontend

A124763 Number of non-rises (levels or falls) for compositions in standard order.

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A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

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A156242 Bisection of A054353.

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A350355 Numbers k such that the k-th composition in standard order is up/down.

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A350356 Numbers k such that the k-th composition in standard order is down/up.

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A337481 Number of compositions of n that are neither strictly increasing nor strictly decreasing.

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A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

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A351011 Numbers k such that the k-th composition in standard order has even length and alternately equal and unequal parts, i.e., all run-lengths equal to 2.

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A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

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