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A127321 First 4-dimensional hyper-tetrahedral coordinate; repeat m C(m+3,3) times; 4-D analog of A056556.

%I A127321 #19 Nov 05 2024 12:18:07
%S A127321 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 A127321 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 A127321 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 A127321 First 4-dimensional hyper-tetrahedral coordinate; repeat m C(m+3,3) times; 4-D analog of A056556.
%C A127321 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 A127321 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 A127321 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 A127321 a(n) = floor(sqrt(5/4 + sqrt(24*n+1)) - 3/2). - _Ridouane Oudra_, Oct 21 2021
%F A127321 a(n) = m-2 if n<binomial(m+2,4) and a(n) = m-1 otherwise where m = floor((24*(n+2))^(1/4)). # _Chai Wah Wu_, Nov 04 2024
%e A127321 a(23)=3 because a(A000332(3+3)) = a(A000332(3+4)-1) = 3, so a(15) = a(34) = 3.
%e A127321 Table of A127321, A127322, A127323, A127324:
%e A127321   n W,X,Y,Z
%e A127321   0 0,0,0,0
%e A127321   1 1,0,0,0
%e A127321   2 1,1,0,0
%e A127321   3 1,1,1,0
%e A127321   4 1,1,1,1
%e A127321   5 2,0,0,0
%e A127321   6 2,1,0,0
%e A127321   7 2,1,1,0
%e A127321   8 2,1,1,1
%e A127321   9 2,2,0,0
%e A127321  10 2,2,1,0
%e A127321  11 2,2,1,1
%e A127321  12 2,2,2,0
%e A127321  13 2,2,2,1
%e A127321  14 2,2,2,2
%e A127321  15 3,0,0,0
%e A127321  16 3,1,0,0
%e A127321  17 3,1,1,0
%e A127321  18 3,1,1,1
%e A127321  19 3,2,0,0
%e A127321  20 3,2,1,0
%e A127321  21 3,2,1,1
%e A127321  22 3,2,2,0
%e A127321  23 3,2,2,1
%t A127321 Array[Floor[Sqrt[5/4 + Sqrt[24*# + 1]] - 3/2] &, 105, 0] (* or *)
%t A127321 Flatten@ Array[ConstantArray[#, Binomial[# + 3, 3]] &, 6, 0] (* _Michael De Vlieger_, Oct 21 2021 *)
%o A127321 (Python)
%o A127321 from math import comb
%o A127321 from sympy import integer_nthroot
%o A127321 def A127321(n): return (m:=integer_nthroot(24*(n+2),4)[0]-2)+(n>=comb(m+4,4)) # _Chai Wah Wu_, Nov 04 2024
%Y A127321 Cf. A127322, A127323, A127324, A056556, A056557, A056558, A000332, A000292, A000217.
%K A127321 nonn
%O A127321 0,6
%A A127321 _Graeme McRae_, Jan 10 2007
%E A127321 Name corrected by _Ridouane Oudra_, Oct 21 2021