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.

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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..65.

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.

Formula

T(0,k) = k.

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

Example

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, ...

Crossrefs

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

Sequence in context: A117524 A308513 A045683 * A157531 A155755 A080088

Adjacent sequences: A343028 A343029 A343030 * A343032 A343033 A343034

Keyword

nonn,tabl

Author

Scott R. Shannon, Apr 03 2021

Status

approved