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

Author

Scott R. Shannon, Feb 01 2021

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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