A056557 Second tetrahedral coordinate.

0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5

Offset: 0

Author

Henry Bottomley, Jun 26 2000

Keywords

• nonn

Comments

If {(X,Y,Z)} are triples of nonnegative integers with X >= Y >= Z ordered by X, Y and Z, then X=A056556(n), Y=A056557(n) and Z=A056558(n).

Crossrefs

Cf. A000292, A002262, A003056, A056556, A056558, A056559, A056560.

See also A194847, A194848.

Programs

Python

from math import isqrt, comb
from sympy import integer_nthroot
def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1),3)[0])-(nChai Wah Wu, Nov 04 2024

Formula

a(n) = floor((sqrt(8*(n-A056556(n)*(A056556(n)+1)*(A056556(n)+2)/6)+1)-1)/2) = A003056(n-A000292(A056556(n)-1)).

a(n+1) = 0 if A056556(n) = A056558(n); a(n+1) = a(n)+1 if a(n) = A056558(n); otherwise a(n+1) = a(n). - Graeme McRae, Jan 09 2007