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

1,1

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.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

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

Example

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)

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]]&]

Prog

(Python)
from itertools import count, islice
def A357489_gen(): # generator of terms
    for n in count(1):
        yield from sorted((1<<n-1)+(1<<x+(y:=m//6)-1)+(1<<y-1) for x in range(1, n) if (m:=2*n-5*x)>0 and 6*(n-x)>m and m%6==0)
A357489_list = list(islice(A357489_gen(), 40)) # Chai Wah Wu, Nov 02 2022

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.

Sequence in context: A020237 A217690 A117565 * A274175 A038491 A020265

Adjacent sequences: A357486 A357487 A357488 * A357490 A357491 A357492

Keyword

nonn

Author

Gus Wiseman, Nov 02 2022

Extensions

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

Status

approved