A356647 Concatenation of runs {y..x} for each x>=1, using each y from 1 to x before moving on to the next value for x.

1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 1, 2, 3, 4, 2, 3, 4, 3, 4, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 3, 4, 5, 4, 5, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 3, 4, 5, 6, 4, 5, 6, 5, 6, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 4, 5, 6, 7, 5, 6, 7, 6, 7, 7, 1, 2, 3

Offset

1,3

Comments

Alternate definition: Flattened list of all suffixes (ordered longest to shortest) of the list of all prefixes (ordered shortest to longest) of the list of positive integers. A prefix here is defined as any contiguous sublist of a list which includes the first element, and a suffix as any contiguous sublist of a list which includes the last element.

Also, concatenation of runs A002260(n)..A002024(n) for each n>=1.

Links

Table of n, a(n) for n=1..87.

The Nineteenth Byte Stack Exchange chat room, Message regarding this sequence. Some replies are programs to generate terms of the sequence.

Formula

a(n) = A000120(A087118(n + 1)). - Thomas Scheuerle, Feb 16 2023

a(n*(n^2 + 5)/6) = a(A004006(n)) = n. This is the earliest appearance of n. - Thomas Scheuerle, Sep 30 2022

Example

1
1 2
_ 2
1 2 3
_ 2 3
_ _ 3
1 2 3 4
_ 2 3 4
_ _ 3 4
_ _ _ 4...

Mathematica

Print @ Flatten @ (Reverse@FoldList[Join[#2, #]&, {#}&/@Reverse@#]& /@ FoldList[Join, Table[{n}, {n, 1, 10}]])

Prog

(JavaScript) a=n=>{for(let i=1; ++i; ){for(let j=0; ++j<i; ){for(let k=j; k<i; k++){if(!--n){return k}}}}}
(MATLAB)
function a = A356647( max_x )
    a = cell2mat(arrayfun(@(x)(cell2mat(arrayfun(@(y)([y:x]), [1:x], 'UniformOutput', false))) ...
        , [1:max_x], 'UniformOutput', false));
end % Thomas Scheuerle, Sep 30 2022
(Python)
from itertools import count, islice
def agen(): # generator of terms
    for k in count(1):
        for j in range(1, k+1):
            yield from range(j, k+1)
print(list(islice(agen(), 87))) # Michael S. Branicky, Oct 11 2022

Crossrefs

Cf. A000120, A004006, A087118.

Sequence in context: A316845 A120481 A369067 * A219644 A193676 A029291

Adjacent sequences: A356644 A356645 A356646 * A356648 A356649 A356650

Keyword

nonn,easy

Author

Jonathan Kal-El Peréz, Aug 19 2022

Status

approved