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

0,6

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.

Links

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

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 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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A329199 A064601 A262303 * A111897 A185711 A185712

Adjacent sequences: A127318 A127319 A127320 * A127322 A127323 A127324

Keyword

nonn

Author

Graeme McRae, Jan 10 2007

Extensions

Name corrected by Ridouane Oudra, Oct 21 2021

Status

approved