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.

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

Author

Gus Wiseman, Feb 03 2022

Keywords

• nonn

Comments

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.

Examples

			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)
		

Crossrefs

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.

Programs

Mathematica

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