A127323 -id:A127323 - cp's OEIS frontend

A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Alternatively, write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives k values. Each n >= 0 has a unique representation as n = C(i,4)+C(j,3)+C(k,2)+C(l.1) with i>j>k>l>=0. This is the combinatorial number system of degree t = 4, where we get [A194882, A194883, A194884, A127324].
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.
This is a 'Matryoshka doll' sequence with alpha=0 (cf. A055462 and A000332), seq(seq(seq(seq(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..4). - Peter Luschny, Jul 14 2009

			See A127321 for a table of A127321, A127322, A127323, A127324
See A127327 for a table of A127324, A127325, A127326, A127327

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A127321, A127322, A127323, A056556, A056557, A056558, A000332, A000292, A000217.

Cf. A002262, A056558.

import Data.List (inits)
a127324 n = a127324_list !! n
a127324_list = concatMap (concatMap concat .
               inits . inits . enumFromTo 0) $ enumFrom 0
-- Reinhard Zumkeller, Jun 01 2015

seq(seq(seq(seq(i,i=0..k),k=0..n),n=0..m),m=0..5); # Peter Luschny, Sep 22 2011

Table[i, {m, 0, 5}, {k, 0, m}, {j, 0, k}, {i, 0, j}] // Flatten  (* Robert G. Wilson v, Sep 27 2011 *)

For W>=X>=Y>=Z>=0, a(A000332(W+3)+A000292(X)+A000217(Y)+Z) = Z A127322(n+1) = A127321(n)==A127324(n) ? 0 : A127322(n)==A127324(n) ? 0 : A127323(n)==A127324(n) ? 0 : A127324(n)+1

A127322 Second 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056557.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 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, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 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

Offset: 0

Graeme McRae, Jan 10 2007

nonn

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 analog of the three-dimensional A056556, A056557 and A056558.

			a(23)=2 because a(A000332(2+3)+A000292(2)) = a(A000332(2+3)+A000292(3)-1) = 2, so a(19) = a(24) = 2.
See A127321 for a table of A127321, A127322, A127323, A127324.

Cf. A127321, A127323, A127324, A056556, A056557, A056558, A000332, A000292, A000217.

For W>=X>=0, a(A000332(W+3)+A000292(X)) = a(A000332(W+3)+A000292(X+1)-1) = X A127322(n+1) = A127321(n)==A127324(n) ? 0 : A127322(n)==A127324(n) ? A127322(n)+1 : A127322(n)

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

Original entry on oeis.org

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

Graeme McRae, Jan 10 2007

nonn

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.

			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

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

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

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 *)

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

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

Name corrected by Ridouane Oudra, Oct 21 2021

A194884 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives k values.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 3, 3, 1, 1, 2, 2, 1, 2, 2, 3, 3, 3, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 1, 1, 2, 2, 1, 2, 2, 3, 3, 3, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 1, 1, 2, 2, 1, 2, 2, 3, 3, 3, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5

Offset: 0

N. J. A. Sloane, Sep 04 2011

nonn

Each n >= 0 has a unique representation as n = C(i,4)+C(j,3)+C(k,2)+C(l.1) with i>j>k>l>=0. This is the combinatorial number system of degree t = 4, where we get [A194882, A194883, A194884, A127324]. For degree t = 3 see A194847.

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

Equals A127323 + 1.

Cf. A194882-A194884, A127324, A194885; A194847, A194848, A056558, A194849.

A127325 Hypertetrahedron with T(W,X,Y,Z) = Y - Z.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Together with A127324, A127326 and A127327 might enable reading "by antidiagonals" of hypercube arrays as 4-dimensional analog of A056558, A056560 and A056559 with cubical arrays.

			a(23)=1 because A127323(23) - A127324(23) = 1.
See A127327 for a table of A127324, A127325, A127326, A127327.

Cf. A127321, A127322, A127323, A127324, A127326, A127327.

Cf. A056558, A056560, A056559.

a(n) = A127323(n) - A127324(n).

A127326 Hypertetrahedron with T(W,X,Y,Z) = X - Y.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Together with A127324, A127325 and A127327 might enable reading "by antidiagonals" of hypercube arrays as 4-dimensional analog of A056558, A056560 and A056559 with cubical arrays.

			a(23)=0 because A127322(23) - A127323(23) = 0.
See A127327 for a table of A127324, A127325, A127326, A127327.

Cf. A127322, A127323, A127324, A127325, A127327.

Cf. A056558, A056560, A056559.

a(n) = A127322(n) - A127323(n).

A127327 Hypertetrahedron with T(W,X,Y,Z) = W - X.

Original entry on oeis.org

0, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Together with A127324, A127325 and A127326 might enable reading "by antidiagonals" of hypercube arrays as 4-dimensional analog of A056558, A056560 and A056559 with cubical arrays.

			a(23)=1 because A127321(23) - A127322(23) = 1.
Table of A127324, A127325, A127326, A127327:
   n w,x,y,z
   0 0,0,0,0
   1 0,0,0,1
   2 0,0,1,0
   3 0,1,0,0
   4 1,0,0,0
   5 0,0,0,2
   6 0,0,1,1
   7 0,1,0,1
   8 1,0,0,1
   9 0,0,2,0
  10 0,1,1,0
  11 1,0,1,0
  12 0,2,0,0
  13 1,1,0,0
  14 2,0,0,0
  15 0,0,0,3
  16 0,0,1,2
  17 0,1,0,2
  18 1,0,0,2
  19 0,0,2,1
  20 0,1,1,1
  21 1,0,1,1
  22 0,2,0,1
  23 1,1,0,1

Cf. A127321, A127322, A127323, A127324, A127325, A127326.

Cf. A056558, A056560, A056559.

a(n) = A127321(n) - A127322(n).

A194885 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; let L[n] = [i,j,k,l]; sequence gives list of quadruples L[n], n >= 0.

Original entry on oeis.org

3, 2, 1, 0, 4, 2, 1, 0, 4, 3, 1, 0, 4, 3, 2, 0, 4, 3, 2, 1, 5, 2, 1, 0, 5, 3, 1, 0, 5, 3, 2, 0, 5, 3, 2, 1, 5, 4, 1, 0, 5, 4, 2, 0, 5, 4, 2, 1, 5, 4, 3, 0, 5, 4, 3, 1, 5, 4, 3, 2, 6, 2, 1, 0, 6, 3, 1, 0, 6, 3, 2, 0, 6, 3, 2, 1, 6, 4, 1, 0, 6, 4, 2, 0, 6, 4, 2, 1, 6, 4, 3, 0, 6, 4, 3, 1, 6, 4, 3, 2, 6, 5, 1, 0, 6, 5, 2, 0, 6, 5, 2, 1, 6, 5, 3, 0, 6, 5, 3, 1, 6

Offset: 0

N. J. A. Sloane, Sep 04 2011

Each n >= 0 has a unique representation as n = C(i,4)+C(j,3)+C(k,2)+C(l.1) with i>j>k>l>=0. This is the combinatorial number system of degree t = 4, where we get [A194882, A194883, A194884, A127324]. For degree t = 3 see A194847.

			List of quadruples begins:
[3, 2, 1, 0]
[4, 2, 1, 0]
[4, 3, 1, 0]
[4, 3, 2, 0]
[4, 3, 2, 1]
[5, 2, 1, 0]
[5, 3, 1, 0]
[5, 3, 2, 0]
[5, 3, 2, 1]
[5, 4, 1, 0]
[5, 4, 2, 0]
...

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

The four columns are [A194882, A194883, A194885, A127324], or equivalently [A127321+3, A127322+2, A127323+1, A127324].

cp's OEIS Frontend

A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A127322 Second 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056557.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A194884 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives k values.

Views

Author

Keywords

Comments

References

Crossrefs

A127325 Hypertetrahedron with T(W,X,Y,Z) = Y - Z.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A127326 Hypertetrahedron with T(W,X,Y,Z) = X - Y.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A127327 Hypertetrahedron with T(W,X,Y,Z) = W - X.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A194885 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; let L[n] = [i,j,k,l]; sequence gives list of quadruples L[n], n >= 0.

Views

Author

Keywords

Comments

Examples

References

Crossrefs