A358333 By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.

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

Offset: 0

Author

Gus Wiseman, Nov 10 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

			Composition 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 5.
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

See link for sequences related to standard compositions (A066099).

Dominates A000120.

Row-lengths of A357135, which is ranked by A357134.

A related sequence is A358330.

Cf. A001511, A029931, A048896, A058891, A070939, A096111, A333766, A357137, A357139, A357186, A357187.

Programs

Mathematica

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

Formula

Sum of A000120 over row n of A066099.