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A342819
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Table read by ascending antidiagonals: T(k, n) is the number of distinct values of the magic constant in a perimeter-magic k-gon of order n.
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6
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4, 4, 7, 6, 9, 10, 6, 11, 12, 13, 8, 13, 16, 17, 16, 8, 15, 18, 21, 20, 19, 10, 17, 22, 25, 26, 25, 22, 10, 19, 24, 29, 30, 31, 28, 25, 12, 21, 28, 33, 36, 37, 36, 33, 28, 12, 23, 30, 37, 40, 43, 42, 41, 36, 31, 14, 25, 34, 41, 46, 49, 50, 49, 46, 41, 34, 14, 27, 36, 45, 50, 55, 56, 57, 54, 51, 44, 37
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OFFSET
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3,1
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LINKS
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Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10-13).
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FORMULA
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O.g.f.: (1 - y + 2*x*(y^2 + y - 1) + x^2*(4*y^2 + y - 3))/((1 - x)^2*(1 + x)*(1 - y)^2*(1 + y)).
E.g.f.: (1 + x*(y - 2))*cosh(x + y) + cosh(y)*sinh(x) + x*(y - 2)*sinh(x + y).
T(k, n) = k*(n - 2) + ((k mod 2) - 1)*(n mod 2) + 1.
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EXAMPLE
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The table begins:
k\n| 3 4 5 6 7 ...
---+-------------------
3 | 4 7 10 13 16 ...
4 | 4 9 12 17 20 ...
5 | 6 11 16 21 26 ...
6 | 6 13 18 25 30 ...
7 | 8 15 22 29 36 ...
...
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MATHEMATICA
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T[k_, n_]:=k(n-2)+(Mod[k, 2]-1)Mod[n, 2]+1; Table[T[k+3-n, n], {k, 3, 14}, {n, 3, k}]//Flatten
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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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