A057555 -id:A057555 - 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.

A057559 Lexicographic ordering of NxNxNxN, where N={1,2,3,...}.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 07 2000

nonn

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

Cf. A057554, A057555, A057556, A057557, A057558, A186006.

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1]; Flatten@lexicographicLattice[{4,4}]
(* by Peter J. C. Moses, Feb 10 2011 *)

Extended by Clark Kimberling, Feb 10 2011

A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 15, 17, 25, 29, 31, 33, 49, 57, 61, 63, 65, 97, 113, 121, 125, 127, 129, 193, 225, 241, 249, 253, 255, 257, 385, 449, 481, 497, 505, 509, 511, 513, 769, 897, 961, 993, 1009, 1017, 1021, 1023, 1025, 1537, 1793, 1921, 1985, 2017, 2033, 2041, 2045, 2047

Offset: 1

Brad Clardy, Apr 01 2013

The table is constructed so that row labels are 2^n - 1, and column labels are 2^n. The body of the table is the row*col + 1. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner.
All of these numbers have the following property:
   let m be a member of A(n),
   if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then
   the differences between successive members of B(n) is a repeating series
   of 1's with the last difference in the pattern m. The number of ones in
   the pattern is 2^j - 1, where j is the column index.
As an example consider A(4) which is 9,
   the sequence B(n) where i XOR 8 = i - 8 starts as:
   8, 9, 10, 11, 12, 13, 14, 15, 24... (A115419)
   with successive differences of:
   1, 1, 1, 1, 1, 1, 1, 9.
The main diagonal is the 6th cyclotomic polynomial evaluated at powers of two (A020515).
The formula for diagonals above the main diagonal
  2^(2*n+1) - 2^(n + (a+1)/2) + 1 n>=(a+1)/2 a=odd number above diagonal
  2^(2*n) - 2^(n  + (b/2)) + 1    n>=(b/2)+1 b=even number above diagonal
The formulas for diagonals below the main diagonal
  2^(2*n+1) - 2^(n + 1 -(a+1)/2) + 1 n>=(a+1)/2 a=odd number below diagonal
  2^(2*n) - 2^(n - (b/2)) + 1       n>=(b/2)+1 b=even number below diagonal
Primes of this sequence are in A152449.

			Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
  +1  |    2    4     8    16    32     64    128    256     512    1024 ...
  ----|-----------------------------------------------------------------
  1   |    3    5     9    17    33     65    129    257     513    1025
  3   |    7   13    25    49    97    193    385    769    1537    3073
  7   |   15   29    57   113   225    449    897   1793    3585    7169
  15  |   31   61   121   241   481    961   1921   3841    7681   15361
  31  |   63  125   249   497   993   1985   3969   7937   15873   31745
  63  |  127  253   505  1009  2017   4033   8065  16129   32257   64513
  127 |  255  509  1017  2033  4065   8129  16257  32513   65025  130049
  255 |  511 1021  2041  4081  8161  16321  32641  65281  130561  261121
  511 | 1023 2045  4089  8177 16353  32705  65409 130817  261633  523265
  1023| 2047 4093  8185 16369 32737  65473 130945 261889  523777 1047553
  ...

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A081118, A152449 (primes), A057555 (lexicographic ordering), A115419 (example).
Rows: A000051(i=1), A181565(2), A083686(3), A195744(4), A206371(5), A196657(6).
Cols: A000225(j=1), A036563(2), A048490(3), A176303 (7 offset of 8).
Diagonals: A020515 (main), A092440, A060867 (above), A134169 (below).

//program generates values in a table form
for i:=1 to 10 do
    m:=2^i - 1;
    m,[ m*2^n +1 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(i-j) + 1;
       if IsPrime(k) then k,"*";
          else k;
       end if;;
    end for;
end for;

Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* Paolo Xausa, Apr 02 2024 *)

a(n) = (2^(A057555(2*n-1)) - 1)*2^(A057555(2*n)) + 1 for n>=1. [corrected by Jason Yuen, Feb 22 2025]

a(n) = A081118(n)+2; a(n)=(2^i-1)*2^j+1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

A057554 Lexicographic ordering of MxM, where M={0,1,2,...}.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 2, 0, 0, 3, 1, 2, 2, 1, 3, 0, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0, 0, 11, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11, 0

Offset: 1

Clark Kimberling, Sep 07 2000

nonn

A057555 gives the lexicographic ordering of N x N, where N={1,2,3,...}.

			Flatten the ordered lattice points: (0,0) < (0,1) < (1,0) < (0,2) < (1,1) < ... as 0,0, 0,1, 1,0, 0,2, 1,1, ...

Alois P. Heinz, Table of n, a(n) for n = 1..20022

Cf. A057555, A057556, A057557, A057558, A057559.

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1]; Flatten@lexicographicLattice[{2,12}]-1 (* Peter J. C. Moses, Feb 10 2011 *)

[l for i in range(20) for k in range(i,-1,-1) for l in (i-k, k)] # Nicholas Stefan Georgescu, Oct 10 2023

A186006 Lexicographic ordering of N x N x N x N x N, where N={1,2,3,...}.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 10 2011

nonn

By changing a single number, the Mathematica code suffices for other dimensions: N x N (A057555), N x N x N (A057557), N x N x N x N (A057559), and higher.

			First, list the 5-tuples in lexicographic order: (1,1,1,1,1) < (1,1,1,1,2) < (1,1,1,2,1) < (1,1,2,1,1) < ... < (1,2,2,1,1) < (1,1,3,1,1) < ... Then flatten the list, leaving 1,1,1,1,1, 1,1,1,1,2, 1,1,1,2,1, 1,1,2,1,1, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A057555, A057557, A057559.

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]
Take[Flatten@lexicographicLattice[{5,12}],160]
(* Peter J. C. Moses, Feb 10 2011 *)

A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

Original entry on oeis.org

21, 37, 53, 69, 101, 117, 133, 197, 229, 245, 261, 389, 453, 485, 501, 517, 773, 901, 965, 997, 1013, 1029, 1541, 1797, 1925, 1989, 2021, 2037, 2053, 3077, 3589, 3845, 3973, 4037, 4069, 4085, 4101, 6149, 7173, 7685, 7941, 8069, 8133, 8165, 8181, 8197, 12293, 14341, 15365, 15877, 16133

Offset: 1

Brad Clardy, Apr 16 2013

The table has row labels 2^n - 1 and column labels 2^(m+3). The table entry is row*col + 5. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner. Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
+5  |   16    32    64   128   256    512   1024 ...
----|-------------------------------------------
1   |   21    37    69   133   261    517   1029
3   |   53   101   197   389   773   1541   3077
7   |  117   229   453   901  1797   3589   7173
15  |  245   485   965  1925  3845   7685  15365
31  |  501   997  1989  3973  7941  15877  31749
63  | 1013  2021  4037  8069 16133  32261  64517
127 | 2037  4069  8133 16261 32517  65029 130053
...
All of these numbers have the following property: let m be a member of A(n); if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then the differences between successive members of B(n) is a repeating series
  of 1,1,1,5 ending with 1,1,1 and the last difference in the pattern m. The total number of 1's and 5's in the pattern is 2^(j+2) - 1, where j is the column index.
As an example, consider A(1), which is 21; the sequence B(n) where i XOR 20 = i - 20 starts as 20, 21, 22, 23, 28, 29, 30, 31, 52, ... with successive differences of 1, 1, 1, 5, 1, 1, 1, 21.
for A(2), which is 37, the sequence B(n) where i XOR 36 = i - 36 starts as 36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 100, ... with successive differences of 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 37.

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A057555 (lexicographic ordering).
Rows: A168614(i=1), n>=4.
Cols: A220087(j=2), n>=6.

//program generates values in a table form, row labels of 2^i -1
for i:=1 to 10 do
    m:=2^i - 1;
    m, [ m*2^(n+3) +5 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(3+i-j) + 5;
       if IsPrime(k) then k, "*";
          else k;
       end if;
    end for;
end for;

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 3) + 5 for n>=1.

A181118 Sequencing of all rational numbers p/q > 0 as ordered pairs (p,q). The rational (p,q) occurs as the n-th ordered pair where n=(p+q-1)*(p+q-2)/2+q.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 1, 3, 4, 1, 3, 2, 2, 3, 1, 4, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3

Offset: 1

Frank M Jackson, Oct 04 2010

From L. Edson Jeffery, Dec 17 2011: (Start)
Arrange the ordered pairs in rows
(1,1)
(2,1),(1,2)
(3,1),(2,2),(1,3)
etc., and let the rows be indexed by n=1,2,.... Then the sum of the products of the pairs in row n is equal to A000292(n). For example, for n=3, 3*1+2*2+1*3=A000292(3)=10. (End)

			Triangle begins:
1,1                  : 1/1;
2,1,1,2              : 2/1, 1/2;
3,1,2,2,1,3          : 3/1, 2/2, 1/3;
4,1,3,2,2,3,1,4      : 4/1, 3/2, 2/3, 1/4;
5,1,4,2,3,3,2,4,1,5  : 5/1, 4/2, 3/3, 2/4, 1/5;
...

G. H. Hardy, A Course of Pure Mathematics (1921), p. 1.

Cf. A000292, A057555.

Flatten[Table [{n+1-r, r}, {n, 9}, {r, n}]]
u[x_] := Floor[3/2 + Sqrt[2*x]]; v[x_] := Floor[1/2 + Sqrt[2*x]]; n[x_] := 1 - x + u[x]*(u[x] - 1)/2; k[x_] := x - v[x]*(v[x] - 1)/2; Flatten[Table[{n[m], k[m]}, {m, 45}]] (* L. Edson Jeffery, Jun 20 2015 *)

for(n=1,9,for(r=1,n,print1(n+1-r", "r", "))) \\ Charles R Greathouse IV, Dec 20 2011

Triangle format R(n,m) of ordered pairs (R(n,2r-1), R(n,2r)) with R(n,2r-1)=n+1-r and R(n,2r)=r and generating the rational (n+1-r)/r.

Typo corrected and tabl changed to tabf by Frank M Jackson, Oct 07 2010

A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.

Original entry on oeis.org

11, 19, 27, 35, 51, 59, 67, 99, 115, 123, 131, 195, 227, 243, 251, 259, 387, 451, 483, 499, 507, 515, 771, 899, 963, 995, 1011, 1019, 1027, 1539, 1795, 1923, 1987, 2019, 2035, 2043, 2051, 3075, 3587, 3843, 3971, 4035, 4067, 4083, 4091, 4099, 6147, 7171, 7683, 7939, 8067, 8131, 8163, 8179

Offset: 1

Brad Clardy, Apr 05 2013

The table has row labels 2^n - 1 and column labels 2^(m+2). The table entry is row*col + 3. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner. Using the lexicographic ordering of A057555  the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
+3  |    8    16    32    64   128    256    512 ...
----|-------------------------------------------
1   |   11    19    35    67   131    259    515
3   |   27    51    99   195   387    771   1539
7   |   59   115   227   451   899   1795   3587
15  |  123   243   483   963  1923   3843   7683
31  |  251   499   995  1987  3971   7939  15875
63  |  507  1011  2019  4035  8067  16131  32259
127 | 1019  2035  4067  8131 16259  32515  65027
...
All of these numbers have the following property: let m be a member of A(n); if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then the differences between successive members of B(n) is an alternating series of 1's and 3's with the last difference in the pattern m. The number of alternating 1's and 3's in the pattern is 2^(j+1) - 1, where j is the column index.
As an example consider A(1) which is 11, the sequence B(n) where i XOR 10 = i - 10 starts as 10, 11, 14, 15, 26, 27, 30, 31, 42, ... (A214864) with successive differences of 1, 3, 1, 11.
Main diagonal is A191341, the largest k such that k-1 and k+1 in binary representation have the same number of 1's and 0's

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A057555(lexicographic ordering), A214864(example), A224195.
Rows: A062729(i=1), A147595(2 n>=5), A164285(3 n>=3).
Cols: A168616(j=1 n>=4).
Diagonal: A191341.

//program generates values in a table form,row labels of 2^i -1
for i:=1 to 10 do
    m:=2^i - 1;
    m, [ m*2^n +1 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(2+i-j) + 3;
       if IsPrime(k) then k, "*";
          else k;
       end if;;
    end for;
end for;

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 2) + 3 for n>=1.

cp's OEIS Frontend

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

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A057559 Lexicographic ordering of NxNxNxN, where N={1,2,3,...}.

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A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

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A057554 Lexicographic ordering of MxM, where M={0,1,2,...}.

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A186006 Lexicographic ordering of N x N x N x N x N, where N={1,2,3,...}.

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A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

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A181118 Sequencing of all rational numbers p/q > 0 as ordered pairs (p,q). The rational (p,q) occurs as the n-th ordered pair where n=(p+q-1)*(p+q-2)/2+q.

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A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.

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