A056558 - OEIS

%I #27 Feb 22 2017 13:30:00

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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)

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

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

%H Reinhard Zumkeller, <a href="/A056558/b056558.txt">Table of n, a(n) for n = 0..10000</a>

%F 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).

%F a(n+1) = A056556(n)==a(n) ? 0 : A056557(n)==a(n) ? 0 : a(n)+1. - _Graeme McRae_, Jan 09 2007

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

%p seq(seq(seq(i,i=0..k),k=0..n),n=0..6); # _Peter Luschny_, Sep 22 2011

%t Table[i, {k, 0, 7}, {j, 0, k}, {i, 0, j}] // Flatten  (* _Robert G. Wilson v_, Sep 27 2011 *)

%o (Haskell)

%o import Data.List (inits)

%o a056558 n = a056558_list !! n

%o a056558_list = concatMap (concat . init . inits . enumFromTo 0) [0..]

%o -- _Reinhard Zumkeller_, Jun 01 2015

%o (PARI) T(t,n,k)=k \\ _Charles R Greathouse IV_, Feb 22 2017

%Y 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.

%Y See also A194847, A194848, A194849.

%Y Cf. A002262, A127324, A000217.

%K nonn

%O 0,10

%A _Henry Bottomley_, Jun 26 2000