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A342719 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.

%I A342719 #21 Apr 18 2022 12:29:45
%S A342719 21,36,45,55,78,78,78,120,136,120,105,171,210,210,171,136,231,300,325,
%T A342719 300,231,171,300,406,465,465,406,300,210,378,528,630,666,630,528,378,
%U A342719 253,465,666,820,903,903,820,666,465,300,561,820,1035,1176,1225,1176,1035,820,561
%N A342719 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.
%H A342719 Terrel Trotter, <a href="https://web.archive.org/web/20070106085340/http://www.trottermath.net/simpleops/pmp.html">Perimeter-Magic Polygons</a>, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equation 3).
%F A342719 O.g.f.: (x^2 - 3*x^2*y + x*y^2 + 3*x^2*y^2)/((1 - x)^3*(1 - y)^3).
%F A342719 E.g.f.: exp(x+y)*x*(x - x*y + y^2 + x*y^2)/2.
%F A342719 T(k, n) = (n - 1)*k*((n - 1)*k + 1)/2.
%e A342719 The array begins:
%e A342719 k\n|   3    4    5    6    7 ...
%e A342719 ---+------------------------
%e A342719 3  |  21   45   78  120  171 ...
%e A342719 4  |  36   78  136  210  300 ...
%e A342719 5  |  55  120  210  325  465 ...
%e A342719 6  |  78  171  300  465  666 ...
%e A342719 7  | 105  231  406  630  903 ...
%e A342719 ...
%t A342719 T[k_,n_]:=(n-1)k((n-1)k+1)/2; Table[T[k+3-n,n],{k,3,12},{n,3,k}]//Flatten
%Y A342719 Cf. A014105 (n = 3), A033585 (n = 5), A037270 (1st superdiagonal), A081266 (n = 4), A083374 (1st subdiagonal), A110450 (diagonal), A144312 (n = 6), A144314 (n = 7), A342757, A342758.
%K A342719 nonn,tabl
%O A342719 3,1
%A A342719 _Stefano Spezia_, Mar 19 2021