A238013 List n copies of each k in {1,2,...,n}, for n=1,2,3,...

1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5

Offset: 1

Author

M. F. Hasler, Feb 16 2014

Keywords

• nonn
• tabf

Comments

Consider a sequence of matrices of size 1 X 1, 2 X 2, 3 X 3, etc., then list their elements, read by rows. The present sequence gives the row index of the n-th term of the sequence within "its" matrix, numbered by A074279(n); the column index being given by A121997.

The sequence could be seen as irregular table or triangle with row lengths equal to 1,2,2,3,3,3,4,4,4,4,... = A002024, each row being filled with k's, for the k-th row of given length; see Example.

Examples

			Formatted as table with row lengths given by A002024 = 1, 2,2, 3,3,3, ... the sequence reads:
1  \\  1 copy of "1";
1 1  \\  2 copies of "1", then
2 2  \\  2 copies of "2";
1 1 1  \\  3 copies of "1", then
2 2 2  \\  3 copies of "2", etc.
3 3 3  \\
1 1 1 1  \\ 1st row of length 4 => filled with "1"s
2 2 2 2  \\ 2nd row of length 4 => filled with "2"s
3 3 3 3  \\ 3rd row of length 4 => filled with "3"s
4 4 4 4  \\ etc.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A121997, A237451, A237452, A002024, A074279.

Programs

Maple

seq(seq(i$j,i=1..j),j=1..10); # Robert Israel, May 29 2017

PARI

for(i=1,9,for(j=1,i,for(k=1,i,print1(j","))))

PARI

A238013_vec(N=9)=concat(vector(N, i, concat(vector(i, j, vector(i, k, i))))) \\ Note: this creates a vector; use A238013_vec()[n] to get the n-th term.

Python

from sympy import integer_nthroot
def A238013(n): return (n-1-(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k-1)*((k<<1)-1)//6)//k+1 # Chai Wah Wu, Nov 04 2024

Formula

a(n) = A237452(n)+1. - Antti Karttunen, Feb 17 2014