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 A056558 Third tetrahedral coordinate, i.e., tetrahedron with T(t,n,k)=k; succession of growing finite triangles with increasing values towards bottom right. 21
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 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 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 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 EXAMPLE First triangle: [0]; second triangle: [0; 0 1]; third triangle: [0; 0 1; 0 1 2]; ... 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 *) PROG (Haskell) import Data.List (inits) a056558 n = a056558_list !! n a056558_list = concatMap (concat . init . inits . enumFromTo 0) [0..] -- Reinhard Zumkeller, Jun 01 2015 (PARI) T(t, n, k)=k \\ Charles R Greathouse IV, Feb 22 2017 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. Sequence in context: A255647 A119241 A001878 * A179519 A091979 A228716 Adjacent sequences:  A056555 A056556 A056557 * A056559 A056560 A056561 KEYWORD nonn AUTHOR Henry Bottomley, Jun 26 2000 STATUS approved

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