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A105373
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Square array by antidiagonals of number of straight lines with n points in a k-dimensional hypercube of with n points on each edge.
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1
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1, 1, 1, 1, 6, 1, 1, 28, 8, 1, 1, 120, 49, 10, 1, 1, 496, 272, 76, 12, 1, 1, 2016, 1441, 520, 109, 14, 1, 1, 8128, 7448, 3376, 888, 148, 16, 1, 1, 32640, 37969, 21280, 6841, 1400, 193, 18, 1, 1, 130816, 192032, 131776, 51012, 12496, 2080, 244, 20, 1, 1, 523776
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OFFSET
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1,5
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LINKS
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FORMULA
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T(1, k)=1. For n>1: T(n, k) = ((n+2)^k-n^k)/2 = (n+2)*T(n, k-1)+n^(k-1) = A102728(k, n+1).
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EXAMPLE
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Rows start:
1, 1, 1, 1, 1, 1, ...;
1, 6, 28, 120, 496, 2016, ...;
1, 8, 49, 272, 1441, 7448, ...;
1, 10, 76, 520, 3376, 21280, ...;
1, 12, 109, 888, 6841, 51012, ...;
etc.
T(5,3)=109 because in a 5 X 5 X 5 cube there are 25 columns, 25 linear rows in one direction, 25 linear rows in another direction, 5 short diagonals in each of 6 directions and 4 long diagonals; and 3*25 + 6*5 + 4 = 109.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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