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

0,13

Links

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

Formula

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

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

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

Example

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.

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]

Prog

(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))+2, 3)<<1))-((r<<2)<=(k<<2)*(k+1)+1) if a==1 else (r:=(b:=n//3)-comb((m:=integer_nthroot((n<<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, Nov 23 2024

Crossrefs

Cf. A056556, A056557, A056558, A330709.

See A372667 for the norms of these triples.

Sequence in context: A353501 A353428 A244738 * A215462 A025843 A303906

Adjacent sequences: A331192 A331193 A331194 * A331196 A331197 A331198

Keyword

nonn,easy,tabf

Author

Mehmet A. Ates, Jun 08 2020

Status

approved