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

Author

Jonathan Kal-El Peréz, Aug 19 2022

Keywords

• nonn
• easy

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.

Examples

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

Links

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

Crossrefs

Cf. A000120, A004006, A087118.

Programs

JavaScript

a=n=>{for(let i=1;++i;){for(let j=0;++j
				

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

Mathematica

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

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

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