A357489 Numbers k such that the k-th composition in standard order is a triple (w,x,y) such that 2w = 3x + 4y.

133, 1034, 4113, 8212, 32802, 65576, 131137, 262212, 524368, 1048706, 2097288, 4194464, 4194561, 8388868, 16777488, 33554752, 33554946, 67109384, 134218272, 134218753, 268436096, 268436484, 536871952, 1073742912, 1073743874, 2147484928, 2147485704, 4294969376

Offset: 1

Author

Gus Wiseman, Nov 02 2022

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers summing to n. 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 the corresponding standard compositions begin:
     133: (5,2,1)
    1034: (7,2,2)
    4113: (8,4,1)
    8212: (9,2,3)
   32802: (10,4,2)
   65576: (11,2,4)
  131137: (11,6,1)
  262212: (12,4,3)
  524368: (13,2,5)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

See link for sequences related to standard compositions.

By sum, these triples appear to be counted by A008676.

The unordered version is A358102, counted by A357849.

A011782 counts compositions.

A066099 lists the standard compositions.

Cf. A000120, A029837, A029931, A070939, A133494.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,10000],Length[stc[#]]==3&&2*stc[#][[1]]==3*stc[#][[2]]+4*stc[#][[3]]&]

Python

from itertools import count, islice
def A357489_gen(): # generator of terms
    for n in count(1):
        yield from sorted((1<0 and 6*(n-x)>m and m%6==0)
A357489_list = list(islice(A357489_gen(),40)) # Chai Wah Wu, Nov 02 2022

Extensions

a(10)-a(28) from Chai Wah Wu, Nov 02 2022