A331195 Three-column table read by rows: triples (i,j,k) in order sorted from the left.

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 2, 0, 0, 2, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 2, 2, 2, 3, 0, 0, 3, 1, 0, 3, 1, 1, 3, 2, 0, 3, 2, 1, 3, 2, 2, 3, 3, 0, 3, 3, 1, 3, 3, 2, 3, 3, 3, 4, 0, 0, 4, 1, 0, 4, 1, 1, 4, 2, 0, 4, 2, 1, 4, 2, 2, 4, 3, 0, 4, 3, 1, 4, 3, 2, 4, 3, 3, 4, 4, 0, 4, 4, 1, 4, 4, 2, 4, 4, 3, 4, 4, 4, 5, 0, 0

Offset: 0

Author

Mehmet A. Ates, Jun 08 2020

Keywords

• nonn
• easy
• tabf

Examples

			For n=[0,1,2] to n=[12,13,14], a[n,n+1,n+2] counts up as such: [0,0,0], [1,0,0], [1,1,0], [1,1,1], [2,0,0], etc.
		

Links

• Mehmet A. Ates, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A056556, A056557, A056558, A330709.

See A372667 for the norms of these triples.

Programs

Mathematica

ThreeDVectors = List[];
SeqSize = 10;
For[i = 0, i <= SeqSize, i++,
  For[j = 0, j <= i, j++,
    For[k = 0, k <= j, k++,
      AppendTo[ThreeDVectors, {i, j, k}]
    ]
  ]
];
Flatten[ThreeDVectors]

Python

from math import comb, isqrt
from sympy import integer_nthroot
def A331195(n): return (m:=integer_nthroot((n<<1)+6,3)[0])-(n<3*comb(m+2,3)) if not (a:=n%3) else (k:=isqrt(r:=(b:=n//3)+1-comb((m:=integer_nthroot((n<<1)-1,3)[0])-(b=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Nov 23 2024

Formula

a(3*n) = A056556(n).

a(3*n+1) = A056557(n).

a(3*n+2) = A056558(n).