A341327 -id:A341327 - cp's OEIS frontend

A340974 The sum of the numbers on straight lines of incrementing length n when drawn over numbers of 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. If two or more lines exist with the same sum the one containing the smallest number is chosen.

1, 5, 18, 46, 95, 171, 238, 372, 549, 775, 1056, 1398, 1807, 2289, 2850, 3482, 3940, 4539, 5525, 6384, 7225, 8263, 9159, 10864, 12032, 13881, 15453, 17094, 18862, 20339, 22758, 25122, 27567, 30605, 33060, 36836, 39285, 43277, 45310, 48850, 53337, 56889, 62264, 65812, 72139, 77531, 81325

Offset: 0

Scott R. Shannon, Feb 01 2021

nonn

The upper and left segments of the spiral contain most of the lines, with the bottom segment containing significantly fewer. Up to 500 lines the only two in the right segment are a(1) = 5 and a(3) = 46. It is unknown if any more appear. The list of numbers that are definitely never covered starts 4,8,9,14,15,16. Whether the next lowest are 38,39,40,... or 27,28,29,... is currently unknown as that is dependent on the existence of further vertical or horizontal lines in the right segment.

Up to 500 lines the only occurrence of two lines with the same sum is a(5) = 171. See the examples below. In this instance if the line with the higher numbers is instead chosen then the value for a(6) becomes 273 but otherwise all other lines and sums are identical to the current sequence.

			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
.
a(0) = 1 as a line of length 0 covers the number 1, which is the minimum possible value.
a(1) = 5 as a line of length 1 is drawn over numbers 2 and 3, which sum to 5. This is the minimum possible sum for such a line which does not use the previously covered number 1.
a(2) = 18 as a line of length 2 is drawn over numbers 5,6,7, which sum to 18. This is the minimum possible sum for such a line which does not use the previously covered numbers 1,2,3.
a(5) = 171 as a line of length 5 is drawn over numbers 22,23,24,25,26,51, which sum to 171. A straight line of length 5 can also be drawn over the uncovered numbers 26,27,28,29,30,31 which also sums to 171, but as the former contains 22, the smallest number of these sets, that is the line chosen. This is the only instance in the first 500 lines where two lines exist with the same sum.

Scott R. Shannon, Image of the first 260 lines. The image can be zoomed in to see the numbers of the square spiral. The colors are graduated across the spectrum to show the lines relative length/placement order.

Cf. A341160 (squares), A341327, A174344, A274923, A296030, A275161.

A341160 The sum of the numbers inside the squares of incrementing size n x n when the square spiral is tiled with these squares, where each tile contains numbers which sum to the minimum possible value, and each number on the spiral can only be in one tile.

Original entry on oeis.org

1, 28, 180, 622, 1910, 3880, 8162, 17592, 28600, 45380, 79376, 122592, 174889, 223556, 313350, 393912, 604421, 792202, 1089859, 1410896, 1644223, 2120976, 2923991, 3369408, 4002500, 5136496, 6298670, 7476224, 8323935, 9464220, 10653646, 12985600, 17233062, 20321768, 22053045, 27665722

Offset: 1

Scott R. Shannon, Feb 06 2021

nonn

See A341327 for the list of the spiral numbers not covered by any square in the tiling.

			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
.
a(1) = 1 as a square of size 1 x 1 is placed on the number 1, which is the minimum possible value.
a(2) = 28 as a square of size 2 x 2 is placed such that it covers the numbers 2,3,11,12 which sum to 28. This is the minimum possible sum for such a square which does not use the previously covered number 1.
a(3) = 180 as a square of size 3 x 3 is placed such that it covers numbers 4,5,18,15,16,17,34,35,36 which sum to 180. This is the minimum possible sum for such a square which does not use the previously covered numbers 1,2,3,11,12.

Scott R. Shannon, Image of the first 60 squares. The image can be zoomed in to see the numbers of the square spiral. The colors are graduated across the spectrum to show the squares relative size/placement order.

Cf. A341327 (spiral numbers not covered), A340974 (lines), A174344, A274923, A296030, A275161.

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

Original entry on oeis.org

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.

A341363 Table read by antidiagonals: T(n, k) is the sum of the numbers inside the k-th square of size n X n when the square spiral is tiled with these squares, where each tile contains numbers which sum to the minimum possible value, and each number on the spiral can only be in one tile.

Original entry on oeis.org

1, 2, 10, 3, 48, 45, 4, 60, 276, 136, 5, 68, 321, 928, 325, 6, 80, 368, 1040, 2349, 666, 7, 92, 384, 1168, 2575, 4984, 1225, 8, 100, 429, 1296, 2825, 5382, 9391, 2080, 9, 124, 456, 1388, 3075, 5816, 10030, 16228, 3321, 10, 128, 554, 1656, 3627, 6250, 10718, 17190, 26257, 5050

Offset: 1

Scott R. Shannon, Feb 10 2021

The terms for a given n tend to have larger jumps in value at one more than the square of the odd numbers, i.e., at k = (2*t+1)^2 + 1, t >= 0, due to the previous square filling a grid of squares containing (2*t+1)^2 squares. This forces the next square to move further away from the origin and into spiral arms containing larger numbers.

See A341278 for the smallest spiral number not covered by any square in each n X n tiling.

			The table begins:
     1,     2,     3,     4,     5,     6,     7,     8,     9,     10, ...
    10,    48,    60,    68,    80,    92,   100,   124,   128,    156, ...
    45,   276,   321,   368,   384,   429,   456,   554,   702,    803, ...
   136,   928,  1040,  1168,  1296,  1388,  1656,  1696,  1858,   2876, ...
   325,  2349,  2575,  2825,  3075,  3627,  3935,  4243,  4415,   7740, ...
   666,  4984,  5382,  5816,  6250,  8456,  9188,  9576, 10154,  14204, ...
  1225,  9391, 10030, 10718, 11406, 15006, 16260, 16737, 17627,  27701, ...
  2080, 16228, 17190, 18216, 19242, 24856, 26856, 27392, 28692,  49240, ...
  3321, 26257, 27636, 29096, 30556, 38998, 42010, 42561, 44383,  81527, ...
  5050, 40344, 42246, 44248, 46250, 58560, 62892, 63400, 65870, 127660, ...
  7381, 59459, 62002, 64666, 67330, 84806, 90808, 91201, 94459, 191129, ...
  ...
.
a(2,1) = 10 as the first square of size 2 X 2 is placed such that it covers the numbers 1,2,3,4, which sum to 10. This is the minimum possible sum.
a(2,2) = 48 as the second square of size 2 X 2 is placed such that it covers the numbers 5,6,18,19, which sum to 48. This is the minimum possible sum for such a square which does not use the previously covered numbers 1,2,3,4.
a(2,3) = 60 as the third square of size 2 X 2 is placed such that it covers the numbers 7,8,22,23, which sum to 60. This is the minimum possible sum for such a square which does not use the previously covered numbers 1,2,3,4,5,6,18,19.

Scott R. Shannon, Image for n=2, k = 1..2000. 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 squares relative placement order.
Scott R. Shannon, Image for n=4, k = 1..1000.
Scott R. Shannon, Image for n=5, k = 1..1000.
Scott R. Shannon, Image for n=7, k = 1..1000.
Scott R. Shannon, Image for n=10, k = 1..500.

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

T(1,k) = k.

T(n,1) = n^2*(n^2+1)/2 = A000217(n^2).

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A341160 The sum of the numbers inside the squares of incrementing size n x n when the square spiral is tiled with these squares, where each tile contains numbers which sum to the minimum possible value, and each number on the spiral can only be in one tile.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

A341363 Table read by antidiagonals: T(n, k) is the sum of the numbers inside the k-th square of size n X n when the square spiral is tiled with these squares, where each tile contains numbers which sum to the minimum possible value, and each number on the spiral can only be in one tile.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula