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

3,1

Links

Table of n, a(n) for n=3..80.

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

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.

Example

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 ...
...

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

Crossrefs

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

Sequence in context: A243454 A204008 A237708 * A016711 A023404 A140245

Adjacent sequences: A342816 A342817 A342818 * A342820 A342821 A342822

Keyword

nonn,tabl

Author

Stefano Spezia, Mar 22 2021

Status

approved