A360240 Weakly decreasing triples of positive integers sorted lexicographically and concatenated.

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

Offset

1,4

Links

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

Formula

a(n) = A331195(n-1) + 1.

Example

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), ...

Mathematica

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

Prog

(Python)
from math import isqrt, comb
from sympy import integer_nthroot
def A360240(n): return (m:=integer_nthroot((n-1<<1)+6, 3)[0])+(n>3*comb(m+2, 3)) if (a:=n%3)==1 else (k:=isqrt(r:=(b:=(n-1)//3)+1-comb((m:=integer_nthroot((n-1<<1)-1, 3)[0])-(b<comb(m+2, 3))+2, 3)<<1))+((r<<2)>(k<<2)*(k+1)+1) if a==2 else 1+(r:=(b:=(n-1)//3)-comb((m:=integer_nthroot((n-1<<1)-3, 3)[0])+(b>=comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2) # Chai Wah Wu, Jun 07 2025

Crossrefs

The triples have sums A070770.

Positions of first appearances are A158842.

For pairs instead of triples we have A330709 + 1.

The zero-based version is A331195.

- The first part is A360010 = A056556 + 1.

- The second part is A194848 = A056557 + 1.

- The third part is A333516 = A056558 + 1.

Cf. A002024, A003056, A069905.

Sequence in context: A008679 A029435 A089643 * A185090 A115268 A103610

Adjacent sequences: A360237 A360238 A360239 * A360241 A360242 A360243

Keyword

nonn

Author

Gus Wiseman, Feb 11 2023

Status

approved