A056558 -id:A056558 - cp's OEIS frontend

A056560 Tetrahedron with T(t,n,k)=n-k; succession of growing finite triangles with increasing values towards bottom left.

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

Offset: 0

Henry Bottomley, Jun 26 2000

nonn

			First triangle: [0]; second triangle: [0; 1 0]; third triangle: [0; 1 0; 2 1 0]; ...

Together with A056558 and A056559 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analog of A002262 and A025581 with square arrays.

a(n) = A056557(n) - A056558(n).

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)

A127323 Third 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056557.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 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 analogs of the three-dimensional A056556, A056557 and A056558.

			a(23)=2 because a(A000332(2+3)+A000292(2)+A000217(2)) = a(A000332(2+3)+A000292(2)+A000217(2+1)-1) = 2, so a(22) = a(24) = 2.
See A127321 for a table of A127321, A127322, A127323, A127324.

Cf. A127321, A127322, A127324, A056556, A056557, A056558, A000332, A000292, A000217.

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

A331195 Three-column table read by rows: triples (i,j,k) in order sorted from the left.

Original entry on oeis.org

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

Offset: 0

Mehmet A. Ates, Jun 08 2020

			For n=[0,1,2] to n=[12,13,14], a[n,n+1,n+2] counts up as such: [0,0,0], [1,0,0], [1,1,0], [1,1,1], [2,0,0], etc.

Mehmet A. Ates, Table of n, a(n) for n = 0..10000

Cf. A056556, A056557, A056558, A330709.

See A372667 for the norms of these triples.

ThreeDVectors = List[];
SeqSize = 10;
For[i = 0, i <= SeqSize, i++,
  For[j = 0, j <= i, j++,
    For[k = 0, k <= j, k++,
      AppendTo[ThreeDVectors, {i, j, k}]
    ]
  ]
];
Flatten[ThreeDVectors]

from math import comb, isqrt
from sympy import integer_nthroot
def A331195(n): return (m:=integer_nthroot((n<<1)+6,3)[0])-(n<3*comb(m+2,3)) if not (a:=n%3) else (k:=isqrt(r:=(b:=n//3)+1-comb((m:=integer_nthroot((n<<1)-1,3)[0])-(b=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Nov 23 2024

a(3*n) = A056556(n).
a(3*n+1) = A056557(n).
a(3*n+2) = A056558(n).

A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 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, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8

Offset: 1

Gus Wiseman, Feb 11 2023

nonn

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...

For pairs instead of triples we have A002024.
Positions of first appearances are A050407(n+2) = A000292(n)+1.
The zero-based version is A056556.
The triples have sums A070770.
The second instead of first part is A194848.
The third instead of first part is A333516.
Concatenating all the triples gives A360240.
Cf. A000217, A003056, A056557, A056558, A069905, A158842, A331195.

nn=9;First/@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

from math import comb
from sympy import integer_nthroot
def A360010(n): return (m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3)) # Chai Wah Wu, Nov 04 2024

a(n) = A056556(n) + 1 = A331195(3n) + 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Feb 18 2024
a(n) = m+1 if n>binomial(m+2,3) and a(n) = m otherwise where m = floor((6n)^(1/3)). - Chai Wah Wu, Nov 04 2024

A070770 b + c + d where b >= c >= d >= 0 ordered by b then c then d.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 4, 5, 6, 3, 4, 5, 5, 6, 7, 6, 7, 8, 9, 4, 5, 6, 6, 7, 8, 7, 8, 9, 10, 8, 9, 10, 11, 12, 5, 6, 7, 7, 8, 9, 8, 9, 10, 11, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 15, 6, 7, 8, 8, 9, 10, 9, 10, 11, 12, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 16, 12, 13, 14, 15, 16, 17, 18, 7

Offset: 0

Henry Bottomley, May 06 2002

			Triangle begins:
  0,
  ;
  1;
  2, 3;
  ;
  2;
  3, 4;
  4, 5, 6;
  ;
  3;
  4, 5,
  5, 6, 7;
  6, 7, 8, 9;
  ;
  4;
  5, 6;
  6, 7,  8;
  7, 8,  9, 10;
  8, 9, 10, 11, 12;
  ;
  ...

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001477, A051162, A070771, A070772 for similar sequences with different numbers of terms summed.

seq(seq(seq(b+c+d,d=0..c),c=0..b),b=0..10); # Robert Israel, Jun 21 2018

for(x=0,5,for(y=0,x,for(z=0,y,print1(x+y+z", ")))) \\ Charles R Greathouse IV, Sep 17 2015

from math import isqrt, comb
from sympy import integer_nthroot
def A070770(n): return (m:=integer_nthroot(6*(n+1),3)[0])+(a:=n>=comb(m+2,3))+(k:=isqrt(b:=(c:=n+1-comb(m+a+1,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1)+c-2-comb(k+(b>k*(k+1)),2) # Chai Wah Wu, Dec 11 2024

a(n) = A056556(n) + A056557(n) + A056558(n).

A194882 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives i values.

Original entry on oeis.org

3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

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 A127321 + 3.

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

from math import comb
from sympy import integer_nthroot
def A194882(n): return (m:=integer_nthroot(24*(n+2),4)[0]+1)+(n>=comb(m+1,4)) # Chai Wah Wu, Dec 10 2024

a(n) = m if n < binomial(m+1,4) and a(n) = m+1 otherwise where m = 1+floor((24*(n+2))^(1/4)). - Chai Wah Wu, Dec 10 2024

A194883 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives j values.

Original entry on oeis.org

2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

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 A127322 + 2.

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

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.

cp's OEIS Frontend

A056560 Tetrahedron with T(t,n,k)=n-k; succession of growing finite triangles with increasing values towards bottom left.

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

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A127323 Third 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056557.

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

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A331195 Three-column table read by rows: triples (i,j,k) in order sorted from the left.

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A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

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A070770 b + c + d where b >= c >= d >= 0 ordered by b then c then d.

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A194882 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives i values.

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