A342757 Array read by ascending antidiagonals: T(k, n) is the minimum value of the magic constant in a perimeter-magic k-gon of order n.
9, 12, 17, 14, 22, 28, 17, 27, 37, 42, 19, 32, 45, 55, 59, 22, 37, 54, 68, 78, 79, 24, 42, 62, 81, 96, 104, 102, 27, 47, 71, 94, 115, 129, 135, 128, 29, 52, 79, 107, 133, 154, 167, 169, 157, 32, 57, 88, 120, 152, 179, 200, 210, 208, 189, 34, 62, 96, 133, 170, 204, 232, 251, 258, 250, 224
Offset: 3
Examples
The array begins: k\n| 3 4 5 6 7 ... ---+------------------------ 3 | 9 17 28 42 59 ... 4 | 12 22 37 55 78 ... 5 | 14 27 45 68 96 ... 6 | 17 32 54 81 115 ... 7 | 19 37 62 94 133 ... ...
Links
- Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10 and 12).
Programs
-
Mathematica
T[k_,n_]:= ((1-Mod[k,2])Mod[n,2]+k*(n^2-2*n+2)+n)/2; Table[T[k+3-n,n],{k,3,13},{n,3,k}]//Flatten
Formula
G.f.: (x^2*(-3*y^3 + 2*y - 1) - x*(2*y^3 + y^2 - 2*y + 1) + (y - 1)*y)/((x - 1)^2*(x + 1)*(y - 1)^3*(y + 1)).
T(k, n) = (n^2/2 - n + 1)*k + n/2 if n is even or both n and k are odd.
T(k, n) = (n^2/2 - n + 1)*k + (n + 1)/2 if n is odd and k is even.
T(k, n) = ((1 - (k mod 2))*(n mod 2) + k*(n^2 - 2*n + 2) + n)/2.