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EXAMPLE
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Triangle T(n, k) starts:
0 | 0;
1 | 1, 2;
2 | 3, 4, 6;
3 | 6, 7, 9, 12;
4 | 10, 11, 13, 16, 20;
5 | 15, 16, 18, 21, 25, 30;
6 | 21, 22, 24, 27, 31, 36, 42;
7 | 28, 29, 31, 34, 38, 43, 49, 56;
8 | 36, 37, 39, 42, 46, 51, 57, 64, 72;
9 | 45, 46, 48, 51, 55, 60, 66, 73, 81, 90;
10 | 55, 56, 58, 61, 65, 70, 76, 83, 91, 100, 110;
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Start at row 0, column 0 with 0. Go down by adding the column index in step n. At row n, restart the counting and go n steps right by adding the row index in step n, then change direction and go down again by adding the column index. After 3*n steps on this path you are at T(2*n, n) which is 2*triangular(n) + (triangular(2*n) - triangular(n)) = (5*n^2 + 3*n)/2. These are the sliced heptagonal numbers A147875 (see the illustration of Leo Tavares).
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The equation T(n, k) = (n*(n + 1) + k*(k + 1))/2 can be extended to all n, k in ZZ.
[n\k] ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 ...
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[-5] ..., 25, 20, 16, 13, 11, 10, 10, 11, 13, 16, 20, 25, ...
[-4] ..., 21, 16, 12, 9, 7, 6, 6, 7, 9, 12, 16, 21, ...
[-3] ..., 18, 13, 9, 6, 4, 3, 3, 4, 6, 9, 13, 18, ...
[-2] ..., 16, 11, 7, 4, 2, 1, 1, 2, 4, 7, 11, 16, ...
[-1] ..., 15, 10, 6, 3, 1, 0, 0, 1, 3, 6, 10, 15, ...
[ 0] ..., 15, 10, 6, 3, 1, 0, 0, 1, 3, 6, 10, 15, ...
[ 1] ..., 16, 11, 7, 4, 2, 1, 1, 2, 4, 7, 11, 16, ...
[ 2] ..., 18, 13, 9, 6, 4, 3, 3, 4, 6, 9, 13, 18, ...
[ 3] ..., 21, 16, 12, 9, 7, 6, 6, 7, 9, 12, 16, 21, ...
[ 4] ..., 25, 20, 16, 13, 11, 10, 10, 11, 13, 16, 20, 25, ...
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