A353432 Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.

0, 1, 10, 21, 26, 43, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 696, 697, 804, 826, 860, 861, 885, 1082, 1141, 1173, 1210, 1338, 1387, 1392, 1393, 1394, 1396, 1594, 1653, 1700, 1720, 1721, 1882, 2106, 2165, 2186

Offset

1,3

Comments

First differs from A353402 (the non-consecutive version) in lacking 53.

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.

Links

Table of n, a(n) for n=1..50.

Example

The initial terms, their binary expansions, and the corresponding standard compositions:
     0:          0  ()
     1:          1  (1)
    10:       1010  (2,2)
    21:      10101  (2,2,1)
    26:      11010  (1,2,2)
    43:     101011  (2,2,1,1)
    58:     111010  (1,1,2,2)
   107:    1101011  (1,2,2,1,1)
   117:    1110101  (1,1,2,2,1)
   174:   10101110  (2,2,1,1,2)
   186:   10111010  (2,1,1,2,2)
   292:  100100100  (3,3,3)
   314:  100111010  (3,1,1,2,2)
   346:  101011010  (2,2,1,2,2)
   348:  101011100  (2,2,1,1,3)
   349:  101011101  (2,2,1,1,2,1)
   373:  101110101  (2,1,1,2,2,1)
   430:  110101110  (1,2,2,1,1,2)
   442:  110111010  (1,2,1,1,2,2)

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
rorQ[y_]:=Length[y]==0||MemberQ[Join@@Table[Take[y, {i, j}], {i, Length[y]}, {j, i, Length[y]}], Length/@Split[y]];
Select[Range[0, 10000], rorQ[stc[#]]&]

Crossrefs

These compositions are counted by A353392.

This is the consecutive case of A353402, counted by A353390.

The non-consecutive recursive version is A353431, counted by A353391.

The recursive version is A353696, counted by A353430.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists compositions in standard order, rev A228351, run-lens A333769.

A329738 counts uniform compositions, partitions A047966.

Statistics of standard compositions:

- Length is A000120, sum A070939.

- Runs are counted by A124767, distinct A351014.

- Subsequences are counted by A334299, contiguous A124770/A124771.

- Runs-resistance is A333628.

Classes of standard compositions:

- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.

- Runs are A272919, counted by A000005.

- Golomb rulers are A333222, counted by A169942.

- Anti-runs are A333489, counted by A003242.

Cf. A044813, A165413, A181819, A318928, A325702, A325705, A325755, A333224, A333755, A353389, A353393, A353403.

Sequence in context: A364079 A367576 A353402 * A331997 A363222 A242287

Adjacent sequences: A353429 A353430 A353431 * A353433 A353434 A353435

Keyword

nonn

Author

Gus Wiseman, May 16 2022

Status

approved