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

3,1

Links

Table of n, a(n) for n=3..68.

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10 and 12).

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.

Example

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 ...
...

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

Crossrefs

Cf. A016873 (n = 4), A285009 (k = 3), A342719, A342758 (maximum).

Sequence in context: A076674 A176062 A027571 * A154631 A199593 A356842

Adjacent sequences: A342754 A342755 A342756 * A342758 A342759 A342760

Keyword

nonn,tabl

Author

Stefano Spezia, Mar 21 2021

Status

approved