A275901 -id:A275901 - cp's OEIS frontend

A276783 a(n) = A275901(n) + A275902(n).

0, 3, 4, 6, 7, 13, 14, 16, 17, 18, 19, 22, 23, 29, 30, 33, 34, 39, 40, 42, 43, 44, 45, 48, 49, 53, 55, 58, 59, 61, 62, 64, 67, 68, 71, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 94, 95, 100, 102, 105, 107, 110, 112, 113, 115, 118, 119, 121, 122, 124, 126, 127, 131, 134, 135, 137, 138, 142, 143, 146, 147, 152, 154, 157, 158, 160, 161, 163, 165, 166, 168, 171

Offset: 0

N. J. A. Sloane, Oct 03 2016

nonn

Specifies which diagonals the queens in A275901 and A275902 lie on.

Cf A275901, A275902, A276325.

Equals A276324 - 1.

# To get 10000 terms of A275902 (xx), A275901 (yy), A276783 (ss), -A276325 (dd)
M1:=100000; M2:=22000; M3:=10000;
xx:=Array(0..M1,0); yy:=Array(0..M1,0); ss:=Array(0..M1,0); dd:=Array(0..M1,0);
xx[0]:=0; yy[0]:=0; ss[0]:=0; dd[0]:=0;
for n from 1 to M2 do
sw:=-1;
   for s from ss[n-1]+1 to M2 do
      for i from 0 to s do
         x:=s-i; y:=i;
         if not member(x,xx,'p') and
            not member(y,yy,'p') and
            not member(x-y,dd,'p') then sw:=1; break; fi;
      od:  # od i
if sw=1 then break; fi;
   od: # od s
  if sw=-1 then lprint("error, n=",n); break; fi;
xx[n]:=x; yy[n]:=y; ss[n]:=x+y; dd[n]:=x-y;
od: # od n
[seq(xx[i],i=0..M3)]:
[seq(yy[i],i=0..M3)]:
[seq(ss[i],i=0..M3)]:
[seq(dd[i],i=0..M3)]:

A275902 Following the successive antidiagonals in A275895, let the n-th queen appear in square (x(n),y(n)); sequence gives y(n).

Original entry on oeis.org

0, 2, 1, 4, 3, 8, 5, 10, 7, 6, 12, 14, 9, 18, 11, 13, 21, 24, 15, 26, 17, 16, 28, 30, 19, 20, 34, 36, 22, 38, 23, 40, 25, 27, 44, 47, 29, 31, 50, 52, 32, 33, 55, 57, 35, 37, 59, 62, 39, 65, 41, 42, 69, 43, 71, 73, 45, 75, 46, 77, 49, 48, 81, 83, 51, 85, 53, 88, 54, 56, 91, 58, 95, 97, 60, 99, 61, 101

Offset: 0

N. J. A. Sloane, Aug 24 2016

nonn

See A275901 for x(n).
This is a permutation of the nonnegative numbers.
This assumes the indexing starts at 0. See A275899, A275900 if the indexing begins at 1.

N. J. A. Sloane, Table of n, a(n) for n = 0..10386

Cf. A065188, A275895, A275899, A275900, A275901.

See A275899.
# Alternative Maple program from N. J. A. Sloane, Oct 03 2016
# To get 10000 terms of A275902 (xx), A275901 (yy), A276783 (ss), -A276325 (dd)
M1:=100000; M2:=22000; M3:=10000;
xx:=Array(0..M1,0); yy:=Array(0..M1,0); ss:=Array(0..M1,0); dd:=Array(0..M1,0);
xx[0]:=0; yy[0]:=0; ss[0]:=0; dd[0]:=0;
for n from 1 to M2 do
sw:=-1;
   for s from ss[n-1]+1 to M2 do
      for i from 0 to s do
         x:=s-i; y:=i;
         if not member(x,xx,'p') and
            not member(y,yy,'p') and
            not member(x-y,dd,'p') then sw:=1; break; fi;
      od:  # od i
if sw=1 then break; fi;
   od: # od s
  if sw=-1 then lprint("error, n=",n); break; fi;
xx[n]:=x; yy[n]:=y; ss[n]:=x+y; dd[n]:=x-y;
od: # od n
[seq(xx[i],i=0..M3)]:
[seq(yy[i],i=0..M3)]:
[seq(ss[i],i=0..M3)]:
[seq(dd[i],i=0..M3)]:

A276325 Diagonal indices of Greedy Queens (see A065188).

Original entry on oeis.org

0, -1, 2, -2, 1, -3, 4, -4, 3, 6, -5, -6, 5, -7, 8, 7, -8, -9, 10, -10, 9, 12, -11, -12, 11, 13, -13, -14, 15, -15, 16, -16, 17, 14, -17, -18, 19, 18, -19, -20, 21, 22, -21, -22, 23, 20, -23, -24, 24, -25, 25, 26, -26, 27, -27, -28, 29, -29, 30, -30, 28, 31

Offset: 1

Alois P. Heinz, Aug 30 2016

sign

a(n) is the index of the diagonal of the n-th queen. The main diagonal has index 0, upper (lower) diagonals have positive (negative) indices.

			The first queen is in the main diagonal, the second queen is in the first lower diagonal, the third queen is in the second upper diagonal, ... :
:
:  Q\\\\ ...
:  \\\Q\ ...
:  \Q\\\ ...
:  \\\\Q ...
:  \\Q\\ ...
:  \\\\\ ...
:  .....

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A065188, A276324, A275901, A275902.

# Maple program from N. J. A. Sloane, Oct 03 2016
# To get 10000 terms of A275902 (xx), A275901 (yy), A276783 (ss), -A276325 (dd)
M1:=100000; M2:=22000; M3:=10000;
xx:=Array(0..M1,0); yy:=Array(0..M1,0); ss:=Array(0..M1,0); dd:=Array(0..M1,0);
xx[0]:=0; yy[0]:=0; ss[0]:=0; dd[0]:=0;
for n from 1 to M2 do
sw:=-1;
   for s from ss[n-1]+1 to M2 do
      for i from 0 to s do
         x:=s-i; y:=i;
         if not member(x,xx,'p') and
            not member(y,yy,'p') and
            not member(x-y,dd,'p') then sw:=1; break; fi;
      od:  # od i
if sw=1 then break; fi;
   od: # od s
  if sw=-1 then lprint("error, n=",n); break; fi;
xx[n]:=x; yy[n]:=y; ss[n]:=x+y; dd[n]:=x-y;
od: # od n
[seq(xx[i],i=0..M3)]:
[seq(yy[i],i=0..M3)]:
[seq(ss[i],i=0..M3)]:
[seq(dd[i],i=0..M3)]:

Equals A275901 - A275902.

A326757 a(n) is the X-coordinate of the n-th nonattacking queen placed by a greedy algorithm on N^3 (see Comments for details).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Jul 23 2019

nonn

We consider an infinite chessboard on N^3 (the first octant of Z^3) traversed by increasing x+y+z and then increasing x+y and then increasing x and place nonattacking queens as soon as possible; these queens can attack along the 13 axes of rotation of a cube.

This sequence is a 3-dimensional variant of A275901.

			The traversal of N^3 starts:
  X  Y  Z
  -  -  -
  0  0  0
  0  0  1
  0  1  0
  1  0  0
  0  0  2
  0  1  1
  1  0  1
  0  2  0
  1  1  0
  2  0  0
  0  0  3
  0  1  2
  1  0  2
  ...
The first queen is placed at position (0, 0, 0) and attacks every position (m*i, m*j, m*k) with max(i, j, k) = 1 and m > 0.
The second queen is placed at position (0, 1, 2).

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A326757
Rémy Sigrist, Interactive scatterplot of the first 25000 queens

See A326758 and A326759 for the Y- and Z- coordinates, respectively.

Cf. A275901.

PARI
```
See Links section.
```

A275899 Following the successive antidiagonals in A065188, let the n-th queen appear in square (x(n),y(n)); sequence gives x(n).

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 10, 7, 11, 13, 8, 9, 15, 12, 20, 21, 14, 16, 26, 17, 27, 29, 18, 19, 31, 34, 22, 23, 38, 24, 40, 25, 43, 42, 28, 30, 49, 50, 32, 33, 54, 56, 35, 36, 59, 58, 37, 39, 64, 41, 67, 69, 44, 71, 45, 46, 75, 47, 77, 48, 78, 80, 51, 52, 85, 53, 86, 55, 90, 91, 57, 95, 60, 61, 99, 62, 101, 63

Offset: 1

N. J. A. Sloane, Aug 24 2016

nonn

See A275900 for y(n).
This is a permutation of the natural numbers.
This assumes the indexing starts at 1. See A275901, A275902 if the indexing begins at 0.

N. J. A. Sloane, Table of n, a(n) for n = 1..7500

Cf. A065188, A275900, A275901, A275902.

# To get the coordinates of queens in order of appearance; b8[] has list of terms of A065188
M:=7500; c1:=[]; c2:=[];
t1:=[seq(n+b8[n],n=1..M)];
t2:=sort(t1);
for n from 1 to M do
i:=t2[n]; member(i,t1,'j');
c1:=[op(c1),j]; c2:=[op(c2),b8[j]];
od:
c3:=map(x->x-1,c1):
c4:=map(x->x-1,c2):
[seq(c1[n],n=1..80)]; # A275899
[seq(c2[n],n=1..80)]; # A275900
[seq(c3[n],n=1..80)]; @ A275901
[seq(c4[n],n=1..80)]; @ A275902

A275900 Following the successive antidiagonals in A065188, let the n-th queen appear in square (x(n),y(n)); sequence gives y(n).

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 6, 11, 8, 7, 13, 15, 10, 19, 12, 14, 22, 25, 16, 27, 18, 17, 29, 31, 20, 21, 35, 37, 23, 39, 24, 41, 26, 28, 45, 48, 30, 32, 51, 53, 33, 34, 56, 58, 36, 38, 60, 63, 40, 66, 42, 43, 70, 44, 72, 74, 46, 76, 47, 78, 50, 49, 82, 84, 52, 86, 54, 89, 55, 57, 92, 59, 96, 98, 61, 100, 62, 102

Offset: 1

N. J. A. Sloane, Aug 24 2016

nonn

See A275899 for x(n).
This is a permutation of the natural numbers.
This assumes the indexing starts at 1. See A275901, A275902 if the indexing begins at 0.

N. J. A. Sloane, Table of n, a(n) for n = 1..7500

Cf. A065188, A275899, A275901, A275902.

Maple
```
See A275899.
```

A276784 Complement of A276783.

Original entry on oeis.org

1, 2, 5, 8, 9, 10, 11, 12, 15, 20, 21, 24, 25, 26, 27, 28, 31, 32, 35, 36, 37, 38, 41, 46, 47, 50, 51, 52, 54, 56, 57, 60, 63, 65, 66, 69, 70, 72, 73, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 96, 97, 98, 99, 101, 103, 104, 106, 108, 109, 111, 114, 116, 117, 120, 123, 125, 128, 129, 130, 132, 133, 136, 139, 140, 141, 144, 145, 148, 149, 150, 151, 153, 155

Offset: 1

N. J. A. Sloane, Oct 03 2016

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..11609

Cf. A275901, A275902, A276783.

cp's OEIS Frontend

A276783 a(n) = A275901(n) + A275902(n).

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A275902 Following the successive antidiagonals in A275895, let the n-th queen appear in square (x(n),y(n)); sequence gives y(n).

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Maple

A276325 Diagonal indices of Greedy Queens (see A065188).

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Formula

A326757 a(n) is the X-coordinate of the n-th nonattacking queen placed by a greedy algorithm on N^3 (see Comments for details).

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PARI

A275899 Following the successive antidiagonals in A065188, let the n-th queen appear in square (x(n),y(n)); sequence gives x(n).

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Programs

Maple

A275900 Following the successive antidiagonals in A065188, let the n-th queen appear in square (x(n),y(n)); sequence gives y(n).

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Comments

Links

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Programs

Maple

A276784 Complement of A276783.

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