A127321 - OEIS

%I #19 Nov 05 2024 12:18:07

%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

%F 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 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 *)

%o (Python)

%o from math import comb

%o from sympy import integer_nthroot

%o 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 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