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A342719
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Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.
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2
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21, 36, 45, 55, 78, 78, 78, 120, 136, 120, 105, 171, 210, 210, 171, 136, 231, 300, 325, 300, 231, 171, 300, 406, 465, 465, 406, 300, 210, 378, 528, 630, 666, 630, 528, 378, 253, 465, 666, 820, 903, 903, 820, 666, 465, 300, 561, 820, 1035, 1176, 1225, 1176, 1035, 820, 561
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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 equation 3).
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FORMULA
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O.g.f.: (x^2 - 3*x^2*y + x*y^2 + 3*x^2*y^2)/((1 - x)^3*(1 - y)^3).
E.g.f.: exp(x+y)*x*(x - x*y + y^2 + x*y^2)/2.
T(k, n) = (n - 1)*k*((n - 1)*k + 1)/2.
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EXAMPLE
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The array begins:
k\n| 3 4 5 6 7 ...
---+------------------------
3 | 21 45 78 120 171 ...
4 | 36 78 136 210 300 ...
5 | 55 120 210 325 465 ...
6 | 78 171 300 465 666 ...
7 | 105 231 406 630 903 ...
...
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MATHEMATICA
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T[k_, n_]:=(n-1)k((n-1)k+1)/2; Table[T[k+3-n, n], {k, 3, 12}, {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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