%I #14 Oct 23 2021 03:20:55
%S 0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%T 3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U 4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N First 4-dimensional hyper-tetrahedral coordinate; repeat m C(m+3,3) times; 4-D analog of A056556.
%C 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.
%H Michael De Vlieger, <a href="/A127321/b127321.txt">Table of n, a(n) for n = 0..10625</a>, showing all instances of m=0..21.
%F For W>=0, a(A000332(W+3)) = a(A000332(W+4)-1) = W A127321(n+1) = A127321(n)==A127324(n) ? A127321(n)+1 : A127321(n)
%F a(n) = floor(sqrt(5/4 + sqrt(24*n+1)) - 3/2). - _Ridouane Oudra_, Oct 21 2021
%e a(23)=3 because a(A000332(3+3)) = a(A000332(3+4)-1) = 3, so a(15) = a(34) = 3.
%e Table of A127321, A127322, A127323, A127324:
%e n W,X,Y,Z
%e 0 0,0,0,0
%e 1 1,0,0,0
%e 2 1,1,0,0
%e 3 1,1,1,0
%e 4 1,1,1,1
%e 5 2,0,0,0
%e 6 2,1,0,0
%e 7 2,1,1,0
%e 8 2,1,1,1
%e 9 2,2,0,0
%e 10 2,2,1,0
%e 11 2,2,1,1
%e 12 2,2,2,0
%e 13 2,2,2,1
%e 14 2,2,2,2
%e 15 3,0,0,0
%e 16 3,1,0,0
%e 17 3,1,1,0
%e 18 3,1,1,1
%e 19 3,2,0,0
%e 20 3,2,1,0
%e 21 3,2,1,1
%e 22 3,2,2,0
%e 23 3,2,2,1
%t Array[Floor[Sqrt[5/4 + Sqrt[24*# + 1]] - 3/2] &, 105, 0] (* or *)
%t Flatten@ Array[ConstantArray[#, Binomial[# + 3, 3]] &, 6, 0] (* _Michael De Vlieger_, Oct 21 2021 *)
%Y Cf. A127322, A127323, A127324, A056556, A056557, A056558, A000332, A000292, A000217.
%K nonn
%O 0,6
%A _Graeme McRae_, Jan 10 2007
%E Name corrected by _Ridouane Oudra_, Oct 21 2021