A056558 Third tetrahedral coordinate, i.e., tetrahedron with T(t,n,k)=k; succession of growing finite triangles with increasing values towards bottom right.

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

Offset: 0

Author

Henry Bottomley, Jun 26 2000

Keywords

• nonn

Comments

Alternatively, write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; sequence gives k values. See A194847 for further information about this interpretation.

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).

This is a 'Matryoshka doll' sequence with alpha=0 (cf. A000292 and A000178). - Peter Luschny, Jul 14 2009

Examples

			First triangle: [0]; second triangle: [0; 0 1]; third triangle: [0; 0 1; 0 1 2]; ...
		

References

• D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Together with A056559 and A056560 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analog of A002262 and A025581 with square arrays. Also cf. A000292, A056556, A056557.

See also A194847, A194848, A194849.

Cf. A002262, A127324, A000217.

Programs

Haskell

import Data.List (inits)
a056558 n = a056558_list !! n
a056558_list = concatMap (concat . init . inits . enumFromTo 0) [0..]
-- Reinhard Zumkeller, Jun 01 2015

Maple

seq(seq(seq(i,i=0..k),k=0..n),n=0..6); # Peter Luschny, Sep 22 2011

Mathematica

Table[i, {k, 0, 7}, {j, 0, k}, {i, 0, j}] // Flatten  (* Robert G. Wilson v, Sep 27 2011 *)

PARI

T(t,n,k)=k \\ Charles R Greathouse IV, Feb 22 2017

Python

from math import isqrt, comb
from sympy import integer_nthroot
def A056558(n): return (r:=n-comb((m:=integer_nthroot(6*(n+1),3)[0])+(n>=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Nov 04 2024

Formula

a(n) = n-A056556(n)*(A056556(n)+1)*(A056556(n)+2)/6-A056557(n)*(A056557(n)+1)/2 = n-A000292(A056556(n)-1)-A000217(A056557(n)) = A056557(n)-A056560(n).

a(n+1) = A056556(n)==a(n) ? 0 : A056557(n)==a(n) ? 0 : a(n)+1. - Graeme McRae, Jan 09 2007