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

0,8

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).

Links

Table of n, a(n) for n=0..104.

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 22.

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

Prog

(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])-(n<comb(m+2, 3))+2, 3)<<1))-((r<<2)<=(k<<2)*(k+1)+1) # Chai Wah Wu, Nov 04 2024

Crossrefs

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

See also A194847, A194848.

Sequence in context: A370303 A020943 A300075 * A342624 A302986 A295305

Adjacent sequences: A056554 A056555 A056556 * A056558 A056559 A056560

Keyword

nonn

Author

Henry Bottomley, Jun 26 2000

Status

approved