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

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

Author

Gus Wiseman, Feb 01 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 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)
		

Crossrefs

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.

Programs

Mathematica

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