A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8

Offset: 1

Author

Gus Wiseman, Feb 11 2023

Keywords

• nonn

Examples

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...
		

Crossrefs

For pairs instead of triples we have A002024.

Positions of first appearances are A050407(n+2) = A000292(n)+1.

The zero-based version is A056556.

The triples have sums A070770.

The second instead of first part is A194848.

The third instead of first part is A333516.

Concatenating all the triples gives A360240.

Cf. A000217, A003056, A056557, A056558, A069905, A158842, A331195.

Programs

Mathematica

nn=9;First/@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

Python

from math import comb
from sympy import integer_nthroot
def A360010(n): return (m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3)) # Chai Wah Wu, Nov 04 2024

Formula

a(n) = A056556(n) + 1 = A331195(3n) + 1.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Feb 18 2024

a(n) = m+1 if n>binomial(m+2,3) and a(n) = m otherwise where m = floor((6n)^(1/3)). - Chai Wah Wu, Nov 04 2024