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A056558
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Third tetrahedral co-ordinate, i.e. tetrahedron with T(t,n,k)=k; succession of growing finite triangles with increasing values towards bottom right.
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21
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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
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OFFSET
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0,10
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COMMENTS
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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
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.
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LINKS
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Table of n, a(n) for n=0..104.
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FORMULA
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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
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EXAMPLE
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First triangle: [0]; second triangle: [0; 0 1]; third triangle: [0; 0 1; 0 1 2]; ...
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MAPLE
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seq(seq(seq(i, i=0..k), k=0..n), n=0..6); # Peter Luschny, Sep 22 2011
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MATHEMATICA
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Table[i, {k, 0, 7}, {j, 0, k}, {i, 0, j}] // Flatten (* Robert G. Wilson v, Sep 27 2011 *)
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CROSSREFS
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Together with A056559 and A056560 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analogue of A002262 and A025581 with square arrays. Also cf. A000292, A056556, A056557.
See also A194847, A194848, A194849.
Cf. A002262, A127324.
Sequence in context: A035168 A119241 A001878 * A179519 A091979 A029430
Adjacent sequences: A056555 A056556 A056557 * A056559 A056560 A056561
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley, Jun 26 2000
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STATUS
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approved
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