Evan Robinson - cp's OEIS frontend

User: Evan Robinson

Evan Robinson has authored 1 sequences.

A356356 Triangle of number of rectangles in the interior of the rectangle with vertices (k,0), (0,k), (n,n+k) and (n+k,n), read by rows.

0, 1, 9, 2, 19, 51, 3, 29, 86, 166, 4, 39, 121, 250, 410, 5, 49, 156, 334, 575, 855, 6, 59, 191, 418, 740, 1141, 1589, 7, 69, 226, 502, 905, 1427, 2044, 2716, 8, 79, 261, 586, 1070, 1713, 2499, 3396, 4356, 9, 89, 296, 670, 1235, 1999, 2954, 4076, 5325, 6645

Offset: 1

Evan Robinson, Oct 15 2022

The function of the triangle T(n,k), where n,k > 0, is equal to (n-k+1)*A330805(k-1) - (n-k)*T(k,k-1) + k*(n-k). This is equivalent to saying that this function is (n-k+1) Aztec diamonds (A330805(k-1)) minus the overlaps of those diamonds (two Aztec diamonds of size k-1 overlapped, hence f(k,k-1)) plus (n-k) copies of k extra rectangles. For this last part, the rectangles are of sizes 1 X (2k-1), 3 X (2k-3), 5 X (2k-5), ..., (2k-3) X 3, (2k-1) X 1 and there are (n-k) copies per overlap.
T(n,n) = A330805(n-1).
If n or k <= 0, T(n,k) = 0.
T(n,k) = T(k,k) + (n-k)*A000447(k). That is, incrementing n for fixed k adds a fixed number of new rectangles, equal to A000447(k).
This sequence was prompted by the codegolf.se question linked below, where the problem was to find T(n,k) plus the number of squares and rectangles in an n X k rectangular lattice with diagonals (lines y+a=+-x).

			Triangle T(n,k) begins:
  n\k  1    2    3    4    5    6    7    8    9   10
   1   0
   2   1    9
   3   2   19   51
   4   3   29   86  166
   5   4   39  121  250  410
   6   5   49  156  334  575  855
   7   6   59  191  418  740 1141 1589
   8   7   69  226  502  905 1427 2044 2716
   9   8   79  261  586 1070 1713 2499 3396 4356
  10   9   89  296  670 1235 1999 2954 4076 5325 6645
For n = 7, k = 3, T(n,k) = (7-3+1)*A330805(3-1) - (7-3)*f(3,2) + 3*(7-3) = 5*51 - 4*19 + 3*4 = 191.

Evan Robinson, First 100 rows, flattened
Code Golf Stack Exchange, Related problem of finding the number of squares on a diagonal grid

Cf. A330805, A000447.

function T(n, k)
    (2*(n-k)*(4*k^3-k)+(4*k^4-k^2-3*k))÷6
end

T(n,k) = (n-k+1)*A330805(k-1) - (n-k)*T(k,k-1) + k*(n-k).
T(n,k) = (n-k+1)*(4*k^4-k^2-3*k)/6 - (n-k)*T(k,k-1) + k*(n-k).
T(n,k) = 1/3*(n-k)*(4*k^3-k) + (4*k^4-k^2-3*k)/6.
T(n,k) = (n-k)*A000447(k) + A330805(k-1).
T(n,1) = n-1.
T(n,n) = A330805(n-1).
T(n,n-1) = (4*n^4-8*n^3-n^2+5*n)/6.
T(n,k) = (n-1)*A000447(k) - T(k,k-1).

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User: Evan Robinson

A356356 Triangle of number of rectangles in the interior of the rectangle with vertices (k,0), (0,k), (n,n+k) and (n+k,n), read by rows.

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