A186003 -id:A186003 - cp's OEIS frontend

A057557 Lexicographic ordering of NxNxN, where N={1,2,3,...}.

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

Offset: 1

Clark Kimberling, Sep 07 2000

nonn

			Flatten the list of ordered lattice points, (1,1,1) < (1,1,2) < (1,2,1) < ..., to 1,1,1, 1,1,2, 1,2,1, ...

A057555: ordering of N^2
A057559: ordering of N^4
A186006: ordering of N^5
A186003: distances to the plane x=0
A186004: distances to the plane y=0
A186005: distances to the plane z=0

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

Corrected and extended by Clark Kimberling,, Feb 10 2011.

A186004 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to xz plane).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 9, 13, 14, 8, 12, 17, 24, 25, 10, 16, 23, 29, 40, 41, 11, 19, 28, 39, 46, 62, 63, 15, 22, 32, 45, 61, 69, 91, 92, 18, 27, 38, 50, 68, 90, 99, 128, 129, 20, 31, 44, 60, 74, 98, 127, 137, 174, 175, 21, 34, 49, 67, 89, 105, 136, 173, 184, 230, 231, 26, 37, 53, 73, 97, 126, 144, 183, 229, 241, 297, 298

Offset: 1

Clark Kimberling, Feb 10 2011

Let n=n(i,j,k) be the position of (i,j,k) in the lexicographic ordering A057557 of N X N X N, where N={1,2,3,...}. Row h of A186004 lists those n for which j=n, the distance from (i,j,k) to the xz-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186004 is a permutation of the positive integers.

			Northwest corner:
   1,  2,  4,  5,  8, 10
   3,  6,  9, 12, 16, 19
   7, 13, 17, 23, 28, 32
  14, 24, 29, 39, 45, 50
  25, 40, 46, 61, 68, 74
T(2,2)=6, the position of (1,2,2) in the ordering
(1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A057557, A186003, A186005.

lexicographicLattice[{dim_,maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1];
lexicographicLatticeHeightArray[{dim_,maxHeight_,axis_}]:=Array[Flatten@Position[Map[#[[axis]]&,lexicographicLattice[{dim,maxHeight}]],#]&,maxHeight];
llha=lexicographicLatticeHeightArray[{3,12,2}];
ordering=lexicographicLattice[{2,Length[llha]}];
llha[[#1,#2]]&@@#1&/@ordering
(* Peter J. C. Moses, Feb 15 2011 *)

A186005 Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane).

Original entry on oeis.org

1, 3, 2, 4, 6, 5, 7, 8, 12, 11, 9, 13, 15, 22, 21, 10, 16, 23, 26, 37, 36, 14, 18, 27, 38, 42, 58, 57, 17, 24, 30, 43, 59, 64, 86, 85, 19, 28, 39, 47, 65, 87, 93, 122, 121, 20, 31, 44, 60, 70, 94, 123, 130, 167, 166, 25, 33, 48, 66, 88, 100, 131, 168, 176, 222, 221, 29, 40, 51, 71, 95, 124, 138, 177, 223, 232, 288, 287

Offset: 1

Clark Kimberling, Feb 10 2011

Let n=n(i,j,k) be the position of (i,j,k) in the lexicographic ordering A057557 of N X N X N, where N={1,2,3,...}. Row h of A186005 lists those n for which k=n, the distance from (i,j,k) to the xy-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186005 is a permutation of the positive integers.

			T(2,2)=6, the position of (1,2,2) in the ordering
(1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...
Northwest corner:
   1,  3,  4,  7,  9, 10
   2,  6,  8, 13, 16, 18
   5, 12, 15, 23, 27, 30
  11, 22, 26, 38, 43, 47
  21, 37, 42, 59, 65, 70

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A057557, A186003, A186004.

lexicographicLattice[{dim_,maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1];
lexicographicLatticeHeightArray[{dim_,maxHeight_,axis_}]:=Array[Flatten@Position[Map[#[[axis]]&,lexicographicLattice[{dim,maxHeight}]],#]&,maxHeight];
llha=lexicographicLatticeHeightArray[{3,12,3}];
ordering=lexicographicLattice[{2,Length[llha]}];
llha[[#1,#2]]&@@#1&/@ordering
(* Peter J. C. Moses, Feb 15 2011 *)

cp's OEIS Frontend

A057557 Lexicographic ordering of NxNxN, where N={1,2,3,...}.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A186004 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to xz plane).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A186005 Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica