cp's OEIS Frontend

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.

%I A356356 #51 Dec 25 2022 14:20:37
%S A356356 0,1,9,2,19,51,3,29,86,166,4,39,121,250,410,5,49,156,334,575,855,6,59,
%T A356356 191,418,740,1141,1589,7,69,226,502,905,1427,2044,2716,8,79,261,586,
%U A356356 1070,1713,2499,3396,4356,9,89,296,670,1235,1999,2954,4076,5325,6645
%N 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.
%C A356356 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.
%C A356356 T(n,n) = A330805(n-1).
%C A356356 If n or k <= 0, T(n,k) = 0.
%C A356356 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).
%C A356356 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).
%H A356356 Evan Robinson, <a href="/A356356/b356356.txt">First 100 rows, flattened</a>
%H A356356 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/questions/87560/count-the-rectangles-in-a-diagonal-grid">Related problem of finding the number of squares on a diagonal grid</a>
%F A356356 T(n,k) = (n-k+1)*A330805(k-1) - (n-k)*T(k,k-1) + k*(n-k).
%F A356356 T(n,k) = (n-k+1)*(4*k^4-k^2-3*k)/6 - (n-k)*T(k,k-1) + k*(n-k).
%F A356356 T(n,k) = 1/3*(n-k)*(4*k^3-k) + (4*k^4-k^2-3*k)/6.
%F A356356 T(n,k) = (n-k)*A000447(k) + A330805(k-1).
%F A356356 T(n,1) = n-1.
%F A356356 T(n,n) = A330805(n-1).
%F A356356 T(n,n-1) = (4*n^4-8*n^3-n^2+5*n)/6.
%F A356356 T(n,k) = (n-1)*A000447(k) - T(k,k-1).
%e A356356 Triangle T(n,k) begins:
%e A356356   n\k  1    2    3    4    5    6    7    8    9   10
%e A356356    1   0
%e A356356    2   1    9
%e A356356    3   2   19   51
%e A356356    4   3   29   86  166
%e A356356    5   4   39  121  250  410
%e A356356    6   5   49  156  334  575  855
%e A356356    7   6   59  191  418  740 1141 1589
%e A356356    8   7   69  226  502  905 1427 2044 2716
%e A356356    9   8   79  261  586 1070 1713 2499 3396 4356
%e A356356   10   9   89  296  670 1235 1999 2954 4076 5325 6645
%e A356356 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.
%o A356356 (Julia)
%o A356356 function T(n, k)
%o A356356     (2*(n-k)*(4*k^3-k)+(4*k^4-k^2-3*k))÷6
%o A356356 end
%Y A356356 Cf. A330805, A000447.
%K A356356 nonn,easy,tabl
%O A356356 1,3
%A A356356 _Evan Robinson_, Oct 15 2022