A341363 -id:A341363 - cp's OEIS frontend

A341278 The smallest spiral number not covered by any square in the minimal-sum square spiral tiling by n X n squares in A341363.

67, 173, 25, 30, 42, 56, 72, 90, 110, 132, 156, 182, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 810, 860, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2067, 2159, 2253, 2349, 2447, 2547, 2649, 2753, 2859, 2967, 3077, 3189

Offset: 2

Scott R. Shannon, Feb 08 2021

nonn

The tilings with n=2 and n=3 are the only ones where the smallest uncovered square is not adjacent to the first centrally placed tile. The sequence starts at n=2 as a 1 X 1 square tiling leaves no squares uncovered.

See A341363 for other images with higher numbers of placed tiles.

Scott R. Shannon, Image of the tiling for n=2. The smallest uncovered square is 67. In this and other images the colors are graduated around the spectrum to show the squares relative placement order.
Scott R. Shannon, Image of the tiling for n=3. The smallest uncovered square is 173.
Scott R. Shannon, Image of the tiling for n=4. The smallest uncovered square is 25.
Scott R. Shannon, Image of the tiling for n=5. The smallest uncovered square is 30.
Scott R. Shannon, Image of the tiling for n=6. The smallest uncovered square is 42.

Cf. A341363, A341160, A341327, A340974, A174344, A274923, A296030, A275161.

A343031 Table read by antidiagonals: T(n, k) is the sum of the numbers on the k-th line of length n when these lines are drawn on the square spiral, where each line contains numbers which sum to the minimum possible value, and each number on the spiral can only be in one line.

Original entry on oeis.org

1, 2, 3, 3, 7, 9, 4, 11, 12, 20, 5, 15, 24, 24, 39, 6, 19, 33, 34, 42, 67, 7, 23, 42, 58, 54, 71, 107, 8, 27, 54, 74, 75, 81, 110, 160, 9, 31, 66, 90, 115, 105, 122, 164, 229, 10, 35, 75, 110, 140, 141, 143, 174, 232, 315, 11, 39, 84, 130, 165, 201, 183, 198, 244, 319, 421

Offset: 0

Scott R. Shannon, Apr 03 2021

Lines of length zero (a single point) and one (two points) can cover the entire square spiral without missing any numbers.
For lines with even numbered length the pattern of lines is very regular, with all lines along the spiral lines of the square spiral, and regular triangles of uncovered numbers along the four diagonals of the spiral. See the linked images with even n.
For odd length lines the pattern formed is more random, with some quadrants have regions, or the entire quadrant, with lines that are orthogonal to the spiral lines, and the triangles of uncovered values becomes more random along the spiral diagonals. See the linked images with odd n.
For n>=2 the smallest spiral number that is not covered by any line is n^2+4n+4.

			The square spiral used is:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
The table begins:
    1,   2,   3,   4,   5,   6,   7,   8,    9,   10,   11,   12, ...
    3,   7,  11,  15,  19,  23,  27,  31,   35,   39,   43,   47, ...
    9,  12,  24,  33,  42,  54,  66,  75,   84,   96,  105,  114, ...
   20,  24,  34,  58,  74,  90, 110, 130,  154,  178,  194,  210, ...
   39,  42,  54,  75, 115, 140, 165, 195,  225,  260,  295,  335, ...
   67,  71,  81, 105, 141, 201, 237, 273,  315,  357,  405,  453, ...
  107, 110, 122, 143, 183, 238, 322, 371,  420,  476,  532,  595, ...
  160, 164, 174, 198, 234, 294, 372, 484,  548,  612,  684,  756, ...
  229, 232, 244, 265, 305, 360, 444, 549,  693,  774,  855,  945, ...
  315, 319, 329, 353, 389, 449, 527, 639,  775,  955, 1055, 1155, ...
  421, 424, 436, 457, 497, 552, 636, 741,  885, 1056, 1276, 1397, ...
  548, 552, 562, 586, 622, 682, 760, 872, 1008, 1188, 1398, 1662, ...

Scott R. Shannon, Image for n=1, k=1..4000. The image can be zoomed in to see the numbers of the square spiral. In this and other images the colors are graduated around the spectrum to show the lines relative placement order.
Scott R. Shannon, Image for n=2, k=1..2000.
Scott R. Shannon, Image for n=3, k=1..2000.
Scott R. Shannon, Image for n=4, k=1..2000.
Scott R. Shannon, Image for n=9, k=1..2000.
Scott R. Shannon, Image for n=10, k=1..2000.
Scott R. Shannon, Image for n=17, k=1..2000.
Scott R. Shannon, Image for n=20, k=1..2000.
Scott R. Shannon, Image for n=21, k=1..2000.

Cf. A340974, A341160, A341363, A341278, A341327, A174344, A274923, A296030, A275161.

T(0,k) = k.

T(1,k) = 3 + 4(k-1).

cp's OEIS Frontend

A341278 The smallest spiral number not covered by any square in the minimal-sum square spiral tiling by n X n squares in A341363.

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