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

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, 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, 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

Offset: 0

Author

Graeme McRae, Jan 10 2007

Keywords

• nonn

Comments

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.

Examples

			a(23)=3 because a(A000332(3+3)) = a(A000332(3+4)-1) = 3, so a(15) = a(34) = 3.
Table of A127321, A127322, A127323, A127324:
  n W,X,Y,Z
  0 0,0,0,0
  1 1,0,0,0
  2 1,1,0,0
  3 1,1,1,0
  4 1,1,1,1
  5 2,0,0,0
  6 2,1,0,0
  7 2,1,1,0
  8 2,1,1,1
  9 2,2,0,0
 10 2,2,1,0
 11 2,2,1,1
 12 2,2,2,0
 13 2,2,2,1
 14 2,2,2,2
 15 3,0,0,0
 16 3,1,0,0
 17 3,1,1,0
 18 3,1,1,1
 19 3,2,0,0
 20 3,2,1,0
 21 3,2,1,1
 22 3,2,2,0
 23 3,2,2,1
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10625, showing all instances of m=0..21.

Crossrefs

Cf. A127322, A127323, A127324, A056556, A056557, A056558, A000332, A000292, A000217.

Programs

Mathematica

Array[Floor[Sqrt[5/4 + Sqrt[24*# + 1]] - 3/2] &, 105, 0] (* or *)
Flatten@ Array[ConstantArray[#, Binomial[# + 3, 3]] &, 6, 0] (* Michael De Vlieger, Oct 21 2021 *)

Python

from math import comb
from sympy import integer_nthroot
def A127321(n): return (m:=integer_nthroot(24*(n+2),4)[0]-2)+(n>=comb(m+4,4)) # Chai Wah Wu, Nov 04 2024

Formula

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).

a(n) = floor(sqrt(5/4 + sqrt(24*n+1)) - 3/2). - Ridouane Oudra, Oct 21 2021

a(n) = m-2 if nChai Wah Wu, Nov 04 2024

Extensions

Name corrected by Ridouane Oudra, Oct 21 2021