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.

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

Author

Scott R. Shannon, Feb 10 2021

Keywords

• nonn
• tabl

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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

Formula

T(1,k) = k.

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