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Array read by ascending antidiagonals: T(k, n) is the maximum value of the magic constant in a perimeter-magic k-gon of order n.
5

%I #5 Mar 23 2021 16:18:40

%S 12,15,23,19,30,37,22,37,48,54,26,44,60,71,74,29,51,71,88,97,97,33,58,

%T 83,105,121,128,123,36,65,94,122,144,159,162,152,40,72,106,139,168,

%U 190,202,201,184,43,79,117,156,191,221,241,250,243,219,47,86,129,173,215,252,281,299,303,290,257

%N Array read by ascending antidiagonals: T(k, n) is the maximum value of the magic constant in a perimeter-magic k-gon of order n.

%H 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 equations 11 and 13).

%F G.f.: (- x^2*(2*y^2 + y - 1) - x*(y^2 + 2*y - 1) + (y - 1)*y^2)/((x - 1)^2*(x + 1)*(y - 1)^3*(y + 1)).

%F T(k, n) = (n^2/2 - 1)*k + n/2 if n is even or both n and k are odd.

%F T(k, n) = (n^2/2 - 1)*k + (n - 1)/2 if n is odd and k is even.

%F T(k, n) = (n + k*(n^2 - 2) + ((k mod 2) - 1)*(n mod 2))/2.

%e The array begins:

%e k\n| 3 4 5 6 7 ...

%e ---+---------------------

%e 3 | 12 23 37 54 74 ...

%e 4 | 15 30 48 71 97 ...

%e 5 | 19 37 60 88 121 ...

%e 6 | 22 44 71 105 144 ...

%e 7 | 26 51 83 122 168 ...

%e ...

%t T[k_,n_]:= (n+k(n^2-2)+(Mod[k,2]-1)Mod[n,2])/2; Table[T[k+3-n,n],{k,3,13},{n,3,k}]//Flatten

%Y Cf. A017005 (n = 4), A135503 (diagonal), A341740 (k = 3), A342719, A342757 (minimum).

%K nonn,tabl

%O 3,1

%A _Stefano Spezia_, Mar 21 2021