A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 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, 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

Offset

0,15

Comments

Alternatively, write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives k values. Each n >= 0 has a unique representation as n = C(i,4)+C(j,3)+C(k,2)+C(l.1) with i>j>k>l>=0. This is the combinatorial number system of degree t = 4, where we get [A194882, A194883, A194884, A127324].

If {(W,X,Y,Z)} are 4-tuples of nonnegative integers with W>=X>=Y>=Z ordered by W, X, Y and Z, then W=A127321(n), X=A127322(n), Y=A127323(n) and Z=A127324(n). These sequences are the four-dimensional analogs of the three-dimensional A056556, A056557 and A056558.

This is a 'Matryoshka doll' sequence with alpha=0 (cf. A055462 and A000332), seq(seq(seq(seq(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..4). - 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

For W>=X>=Y>=Z>=0, a(A000332(W+3)+A000292(X)+A000217(Y)+Z) = Z A127322(n+1) = A127321(n)==A127324(n) ? 0 : A127322(n)==A127324(n) ? 0 : A127323(n)==A127324(n) ? 0 : A127324(n)+1

Example

See A127321 for a table of A127321, A127322, A127323, A127324
See A127327 for a table of A127324, A127325, A127326, A127327

Maple

seq(seq(seq(seq(i, i=0..k), k=0..n), n=0..m), m=0..5); # Peter Luschny, Sep 22 2011

Mathematica

Table[i, {m, 0, 5}, {k, 0, m}, {j, 0, k}, {i, 0, j}] // Flatten  (* Robert G. Wilson v, Sep 27 2011 *)

Prog

(Haskell)
import Data.List (inits)
a127324 n = a127324_list !! n
a127324_list = concatMap (concatMap concat .
               inits . inits . enumFromTo 0) $ enumFrom 0
-- Reinhard Zumkeller, Jun 01 2015

Crossrefs

Cf. A127321, A127322, A127323, A056556, A056557, A056558, A000332, A000292, A000217.

Cf. A002262, A056558.

Sequence in context: A328831 A085981 A243224 * A083917 A117974 A369927

Adjacent sequences: A127321 A127322 A127323 * A127325 A127326 A127327

Keyword

nonn

Author

Graeme McRae, Jan 10 2007

Status

approved