A144625 -id:A144625 - cp's OEIS frontend

A144626 Tetrahedron of numbers T(i,j,k) = (i+2j+3k)!/(i!j!k!2^j6^k) read with entries in the order defined in A144625.

1, 1, 1, 1, 1, 3, 3, 4, 10, 10, 1, 6, 15, 15, 10, 60, 105, 70, 280, 280, 1, 10, 45, 105, 105, 20, 210, 840, 1260, 280, 2520, 6300, 2800, 15400, 15400, 1, 15, 105, 420, 945, 945, 35, 560, 3780, 12600, 17325, 840, 12600, 69300, 138600, 15400, 184800, 600600, 200200, 1401400, 1401400

Offset: 0

N. J. A. Sloane, Jan 18 2009, Jan 19 2009

The slice sums are given by A144416.

			The n-th slice of the tetrahedrom consists of the terms T(i,j,k) with i+j+k = n.
Slices 0,1,2,3,4,5 are:
....................1
------------------
....................1
...................1.1
------------------
....................10
...................4.10
..................1.3..3
------------------
...................280
..................70.280
................10.60.105
...............1..6.15..15
------------------
..................15400
................2800.15400
...............280.2520.6300
.............20..210.840.1260
............1..10..45..105.105
------------------
.................1401400
.............200200.1401400
.........15400.184800...600600
......840..12600..69300...138600
...35....560....3780....12600...17325
1......15....105....420......945.....945

Cf. A144416, A144625, A144385.

A144629 Last members of triples listed in A144625.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jan 18 2009

nonn

A144627 Initial members of triples listed in A144625.

Original entry on oeis.org

0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 3, 2, 1, 0, 2, 1, 0, 1, 0, 0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 1, 0, 0, 5, 4, 3, 2, 1, 0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 1, 0, 0, 6, 5, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 1, 0, 0, 7, 6, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0

Offset: 0

N. J. A. Sloane, Jan 18 2009

nonn

for n from 0 to 7 do
for k from 0 to n do for j from 0 to n do for i from 0 to n do if i+j+k=n then lprint(i,j,k); fi;
od: od: od: od:

DeleteCases[Flatten[Table[If[i+j+k==n,i],{n,0,10},{k,0,n},{j,0,n},{i,0,n}]],Null] (* Harvey P. Dale, Mar 27 2012 *)

A144628 Central members of triples listed in A144625.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6

Offset: 0

N. J. A. Sloane, Jan 18 2009

nonn

A057556 Lexicographic ordering of M x M x M, where M={0,1,2,...}.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 07 2000

See A057557 for N x N x N, where N={1,2,3,...}.

The triples are sorted first according to their sum, then lexicographically. - Pontus von Brömssen, Aug 16 2023

			Flatten the list of ordered lattice points, (0,0,0) < (0,0,1) < (0,1,0) < ... to 0,0,0, 0,0,1, 0,1,0, ...
As a three-column array:
  0 0 0
  0 0 1
  0 1 0
  1 0 0
  0 0 2
  0 1 1
  0 2 0
  1 0 1
  1 1 0
  2 0 0
  0 0 3
  0 1 2
  0 2 1
  0 3 0
  1 0 2
  1 1 1
  1 2 0
  2 0 1
  2 1 0
  3 0 0
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..10962

Cf. A057554, A057557.

Cf. A144625 (each triple reversed).

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1]; Flatten@lexicographicLattice[{3,6}]-1
(* Peter J. C. Moses, Feb 10 2011 *)

Extended by Clark Kimberling, Feb 10 2011

A365139 List of free polycubes in binary code (see comments), ordered first by the number of cells, then by the value of the binary code.

Original entry on oeis.org

1, 3, 7, 19, 15, 23, 39, 43, 51, 54, 1043, 31, 47, 55, 59, 87, 118, 173, 179, 182, 199, 230, 1047, 1075, 1078, 2071, 2075, 2149, 2150, 2164, 2214, 2218, 6182, 1049619, 63, 95, 119, 175, 183, 190, 207, 215, 231, 237, 238, 246, 423, 430, 438, 1055, 1079, 1083

Offset: 1

Pontus von Brömssen, Aug 23 2023

The binary code used here is a straight-forward generalization of the binary code in A246521 to d > 2 dimensions. Order the d-tuples of nonnegative integers, first according to their sum, then colexicographically. (For the purposes of this definition, the result will be the same if we use lexicographic order instead.) Label the d-tuples 0, 1, 2, ... in this order. (For d = 3, this is the ordering of triples given by A144625.) Given a d-dimensional polyomino (represented as a finite set of integer d-tuples), consider all the d!*2^d ways of rotating/reflecting it. Translate each such rotation/reflection so that the minimum coordinate is 0 in each dimension, and add the powers of 2 with exponents equal to the labels of the d-tuples of the translation. The binary code of the polyomino (or any finite set of d-tuples) is the minimum of those sums.

Can be read as an irregular triangle, whose n-th row contains A038119(n) terms.

			Consider the pentacube consisting of a straight tricube with two monocubes attached to two adjacent faces of its middle cube. The following table shows the first few triples (with their ordinal number in front), with those triples appearing in the orientation of the pentacube that minimizes the binary code marked with an "X":
  0. 000 X
  1. 100 X
  2. 010
  3. 001
  4. 200 X
  5. 110 X
  6. 020
  7. 101 X
  8. 011
  9. 002
Consequently, the binary code of this pentacube is 2^0+2^1+2^4+2^5+2^7 = 179 = a(19).
As an irregular triangle:
  1;
  3;
  7, 19;
  15, 23, 39, 43, 51, 54, 1043;
  ...

Index entries for sequences related to polyominoes.

Cf. A038119, A144625, A246521 (2 dimensions), A365140 (4 dimensions), A365141 (5 dimensions).

cp's OEIS Frontend

A144626 Tetrahedron of numbers T(i,j,k) = (i+2*j+3*k)!/(i!*j!*k!*2^j*6^k) read with entries in the order defined in A144625.

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A144629 Last members of triples listed in A144625.

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A144627 Initial members of triples listed in A144625.

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A144628 Central members of triples listed in A144625.

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A057556 Lexicographic ordering of M x M x M, where M={0,1,2,...}.

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A365139 List of free polycubes in binary code (see comments), ordered first by the number of cells, then by the value of the binary code.

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A144626 Tetrahedron of numbers T(i,j,k) = (i+2j+3k)!/(i!j!k!2^j6^k) read with entries in the order defined in A144625.