A274917 -id:A274917 - cp's OEIS frontend

A275609 Square spiral in which each new term is the least nonnegative integer distinct from its (already assigned) eight neighbors.

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

Offset: 0

Omar E. Pol, Nov 14 2016

nonn

"Neighbor" here means the eight cells surrounding a cell (cells that are a chess king's move away). The number assigned to a cell is the mex of the numbers that have already been assigned to any of its eight neighbors. - N. J. A. Sloane, Mar 24 2019
The largest element is 4 and it is also the element with lower density in the spiral.
[Proof that 4 is the largest term. When the spiral is being filled in, the maximal number of its neighbors that have already been filled in is four. The mex of four nonnegative numbers is at most 4. QED - N. J. A. Sloane, Mar 24 2019]
For more information see also A307188. - Omar E. Pol, Apr 01 2019

			Illustration of initial terms as a spiral (n = 0..168):
.
.     1 - 2 - 1 - 0 - 4 - 0 - 2 - 0 - 1 - 3 - 1 - 3 - 1
.     |                                               |
.     3   0 - 3 - 2 - 1 - 3 - 1 - 3 - 2 - 0 - 2 - 0   2
.     |   |                                       |   |
.     1   2   1 - 0 - 4 - 0 - 2 - 0 - 1 - 3 - 1   3   1
.     |   |   |                               |   |   |
.     0   4   3   2 - 1 - 3 - 1 - 3 - 2 - 0   2   0   2
.     |   |   |   |                       |   |   |   |
.     3   1   0   4   0 - 2 - 0 - 4 - 1   3   1   3   1
.     |   |   |   |   |               |   |   |   |   |
.     0   2   3   1   3   1 - 3 - 2   0   2   0   2   0
.     |   |   |   |   |   |       |   |   |   |   |   |
.     3   1   0   2   0   2   0 - 1   3   1   3   1   3
.     |   |   |   |   |   |           |   |   |   |   |
.     0   2   3   1   3   1 - 3 - 2 - 0   2   0   2   0
.     |   |   |   |   |                   |   |   |   |
.     3   1   0   2   0 - 2 - 0 - 1 - 3 - 1   3   1   3
.     |   |   |   |                           |   |   |
.     0   2   3   1 - 3 - 1 - 3 - 2 - 0 - 2 - 0   2   0
.     |   |   |                                   |   |
.     3   1   0 - 2 - 0 - 2 - 0 - 1 - 3 - 1 - 3 - 1   3
.     |   |                                           |
.     0   2 - 3 - 1 - 3 - 1 - 3 - 2 - 0 - 2 - 0 - 2 - 0
.     |
.     1 - 4 - 0 - 2 - 0 - 2 - 0 - 1 - 3 - 1 - 3 - 1 - 3
.
a(13) = 4 is the first "4" in the sequence and its four neighbors are 3 (southwest), 2 (south), 0 (southeast) and 1 (east) when a(13) is placed in the spiral.
a(157) = 4 is the 6th "4" in the sequence and it is also the first "4" that is below the NE-SW main diagonal of the spiral (see the second term in the last row of the above diagram).

Alois P. Heinz, Table of n, a(n) for n = 0..100000 (first 5001 terms from N. J. A. Sloane)
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Rémy Sigrist, Colored illustration of the sequence (with cells (x,y) such that -100 <= x <= 100 and -100 <= y <= 100)
N. J. A. Sloane, Central portion of spiral, shown without spaces [The central 0 (at (0,0)) has been changed to an X. The illustration shows cells (x,y) with -35 <= x <= 35, -33 <= y <= 36.]

Cf. A274912, A274917, A274920, A275606, A278354 (number of neighbors).
See A307188-A307192 for the positions of 0,1,2,3,4 respectively.
The eight spokes starting at the origin are A307193 - A307200.

fx:= proc(n) option remember; `if`(n=1, 0, (k->
       fx(n-1)+sin(k*Pi/2))(floor(sqrt(4*(n-2)+1)) mod 4))
     end:
fy:= proc(n) option remember; `if`(n=1, 0, (k->
       fy(n-1)-cos(k*Pi/2))(floor(sqrt(4*(n-2)+1)) mod 4))
     end:
b:= proc() -1 end:
a:= proc(n) option remember; local x, y, s, m;
      x, y:= fx(n+1), fy(n+1);
      if n>0 then a(n-1) fi;
      if b(x, y) >= 0 then b(x, y)
    else s:= {b(x+1, y+1), b(x-1, y-1), b(x+1, y-1), b(x-1, y+1),
              b(x+1, y  ), b(x-1, y  ), b(x  , y+1), b(x  , y-1)};
         for m from 0 while m in s do od;
         b(x, y):= m
      fi
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Mar 29 2019

fx[n_] := fx[n] = If[n == 1, 0, Function[k, fx[n - 1] + Sin[k*Pi/2]][Mod[ Floor[Sqrt[4*(n - 2) + 1]], 4]]];
fy[n_] := fy[n] = If[n == 1, 0, Function[k, fy[n - 1] - Cos[k*Pi/2]][Mod[ Floor[Sqrt[4*(n - 2) + 1]], 4]]];
b[, ] := -1;
a[n_] := a[n] = Module[{x, y, s, m}, {x, y} = {fx[n + 1], fy[n + 1]}; If[n > 0, a[n - 1]]; If [b[x, y] >= 0, b[x, y], s = {b[x + 1, y + 1], b[x - 1, y - 1], b[x + 1, y - 1], b[x - 1, y + 1], b[x + 1, y], b[x - 1, y], b[x, y + 1], b[x, y - 1]}; For[m = 0, MemberQ[s, m], m++]; b[x, y] = m]];
a /@ Range[0, 120] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

a(n) = A274917(n) - 1.

A278180 Square spiral in which each new term is the sum of its two largest neighbors.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 8, 15, 16, 17, 33, 35, 37, 72, 76, 80, 84, 164, 172, 180, 188, 368, 384, 401, 418, 435, 853, 888, 925, 962, 999, 1961, 2037, 2117, 2201, 2285, 2369, 4654, 4826, 5006, 5194, 5382, 5570, 10952, 11336, 11737, 12155, 12590, 13025, 13460, 26485, 27373, 28298, 29260, 30259, 31258, 32257, 63515

Offset: 1

Omar E. Pol, Nov 14 2016

nonn

To evaluate a(n) consider the neighbors of a(n) that are present in the spiral when a(n) should be a new term in the spiral.

For the same idea but for a hexagonal spiral see A278619; and for a right triangle see A278645. It appears that the same idea for an isosceles triangle and also for a square array gives A030237. - Omar E. Pol, Dec 04 2016

			Illustration of initial terms as a square spiral:
.
.          84----80----76-----72----37
.           |                        |
.          164    4-----3-----2     35
.           |     |           |      |
.          172    7     1-----1     33
.           |     |                  |
.          180    8-----15----16----17
.           |
.          188---368---384---401---418
.
a(21) = 188 because the sum of its two largest neighbors is 180 + 8 = 188.
a(22) = 368 because the sum of its two largest neighbors is 180 + 188 = 368.
a(23) = 384 because the sum of its two largest neighbors is 368 + 16 = 384.
a(24) = 401 because the sum of its two largest neighbors is 384 + 17 = 401.
a(25) = 418 because the sum of its two largest neighbors is 401 + 17 = 418.
a(26) = 435 because the sum of its two largest neighbors is 418 + 17 = 435.

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Bitmap illustrating the parity of the first one million terms. (Even and odd numbers are represented with black and white pixels respeectively.)

Cf. A030237, A078510, A141481, A274917, A278181, A278619, A278645.

A278354 Number of neighbors of each new term in a square spiral.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 19 2016

nonn

Here the "neighbors" of a(n) are defined to be the adjacent elements to a(n) in the same row, column or diagonals, that are present in the spiral when a(n) is the new element of the sequence in progress.

For the same idea but for a right triangle see A278317; for an isosceles triangle see A275015; for a square array see A278290; and for a hexagonal spiral see A047931.

			Illustration of initial terms as a spiral (n = 1..169):
.
.     2 - 3 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 2
.     |                                               |
.     4   2 - 3 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 2   3
.     |   |                                       |   |
.     4   4   2 - 3 - 4 - 4 - 4 - 4 - 4 - 4 - 2   3   4
.     |   |   |                               |   |   |
.     4   4   4   2 - 3 - 4 - 4 - 4 - 4 - 2   3   4   4
.     |   |   |   |                       |   |   |   |
.     4   4   4   4   2 - 3 - 4 - 4 - 2   3   4   4   4
.     |   |   |   |   |               |   |   |   |   |
.     4   4   4   4   4   2 - 3 - 2   3   4   4   4   4
.     |   |   |   |   |   |       |   |   |   |   |   |
.     4   4   4   4   4   3   0 - 1   4   4   4   4   4
.     |   |   |   |   |   |           |   |   |   |   |
.     4   4   4   4   3   2 - 4 - 3 - 2   4   4   4   4
.     |   |   |   |   |                   |   |   |   |
.     4   4   4   3   2 - 4 - 4 - 4 - 3 - 2   4   4   4
.     |   |   |   |                           |   |   |
.     4   4   3   2 - 4 - 4 - 4 - 4 - 4 - 3 - 2   4   4
.     |   |   |                                   |   |
.     4   3   2 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 3 - 2   4
.     |   |                                           |
.     3   2 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 3 - 2
.     |
.     2 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 3
.

Cf. A002620, A033638, A047931, A141481, A274917, A275015, A275609, A278180, A278290, A278317.

0,1,seq(op([2,4$floor(i/2),3]),i=0..30); # Robert Israel, Nov 22 2016

From Robert Israel, Nov 22 2016: (Start)
a(n) = 3 if n>=4 is in A002620.
a(n) = 2 if n>=2 is in A033638.
Otherwise, a(n) = 4 if n > 2. (End)

cp's OEIS Frontend

A275609 Square spiral in which each new term is the least nonnegative integer distinct from its (already assigned) eight neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A278180 Square spiral in which each new term is the sum of its two largest neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A278354 Number of neighbors of each new term in a square spiral.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula