A342819 Table read by ascending antidiagonals: T(k, n) is the number of distinct values of the magic constant in a perimeter-magic k-gon of order n.

4, 4, 7, 6, 9, 10, 6, 11, 12, 13, 8, 13, 16, 17, 16, 8, 15, 18, 21, 20, 19, 10, 17, 22, 25, 26, 25, 22, 10, 19, 24, 29, 30, 31, 28, 25, 12, 21, 28, 33, 36, 37, 36, 33, 28, 12, 23, 30, 37, 40, 43, 42, 41, 36, 31, 14, 25, 34, 41, 46, 49, 50, 49, 46, 41, 34, 14, 27, 36, 45, 50, 55, 56, 57, 54, 51, 44, 37

Offset: 3

Author

Stefano Spezia, Mar 22 2021

Keywords

• nonn
• tabl

Examples

			The table begins:
k\n|  3   4   5   6   7 ...
---+-------------------
3  |  4   7  10  13  16 ...
4  |  4   9  12  17  20 ...
5  |  6  11  16  21  26 ...
6  |  6  13  18  25  30 ...
7  |  8  15  22  29  36 ...
...
		

Links

• Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10-13).

Crossrefs

Cf. A005408 (n = 4), A016813 (n = 6), A016921 (n = 8), A017077 (n = 10), A146951 (n = 7), A238290 (n = 9), A342757, A342758.

Programs

Mathematica

T[k_,n_]:=k(n-2)+(Mod[k,2]-1)Mod[n,2]+1;Table[T[k+3-n,n],{k,3,14},{n,3,k}]//Flatten

Formula

O.g.f.: (1 - y + 2*x*(y^2 + y - 1) + x^2*(4*y^2 + y - 3))/((1 - x)^2*(1 + x)*(1 - y)^2*(1 + y)).

E.g.f.: (1 + x*(y - 2))*cosh(x + y) + cosh(y)*sinh(x) + x*(y - 2)*sinh(x + y).

T(k, n) = k*(n - 2) + ((k mod 2) - 1)*(n mod 2) + 1.

T(k, n) = A342758(k, n) - A342757(k, n) + 1.