A380962 -id:A380962 - cp's OEIS frontend

A380964 Perimeter-magic hexagons of order 3 with magic sum n.

9, 48, 150, 494, 1202, 2542, 4635, 9738, 14943, 25917, 41196, 62518, 89657, 139743, 185114, 264483, 363291, 485411, 630099, 862106, 1067459, 1391011, 1771817, 2210554, 2712337, 3461467, 4115434, 5073010, 6165577, 7387876, 8748214, 10655591, 12333486, 14679050, 17281206

Offset: 17

Derek Holton and Alex Holton, Feb 09 2025

nonn

Each side of the hexagon has 3 integers (=the order), 2 of them shared by adjacent sides. All 12 integers on the vertices must be distinct. Solutions obtained by rotations around the 6-fold axis or flips are considered the same/equivalent (bracelet symmetry).

A244879(n-3) counts the perimeter-magic hexagons of order 3 if the 12 integers do not need to be distinct and if solutions by rotations/reflections are considered distinct. - R. J. Mathar, Mar 10 2025

			For magic sum 17, a(17) = 9. One of the hexagons is   5   9   3
                                                    10          8
                                                   2             6
                                                    14          7
                                                      1   12   4

Bert Dobbelaere, Table of n, a(n) for n = 17..100
R. J. Mathar, Generating Perimeter-magic Polygons (C++ source) (2025)

Cf. A380853 (triangles), A380962 (squares), A380963 (pentagons).

More terms from Bert Dobbelaere, Mar 15 2025

A380963 The number of perimeter-magic pentagons of order 3 with magic sum n.

Original entry on oeis.org

1, 9, 33, 75, 233, 374, 742, 1294, 2042, 3029, 4931, 6535, 9507, 13214, 17577, 22762, 31335, 38341, 49660, 62791, 77689, 94239, 119151, 139727, 170553, 204832, 242122, 282811, 340914, 388834, 456668, 530819, 609982, 694982, 810204, 906951, 1038672

Offset: 14

Derek Holton and Alex Holton, Feb 09 2025

nonn

The requirements are that there are 3 integers at each side of the pentagon (2 of them shared by adjacent sides), which sum up to n. All 10 integers on the 5 sides must be distinct. Pentagons obtained by reflections or rotations are considered to be the same.

If the 10 integers do not need to be distinct and if solutions by rotations around the five-fold symmetry axis and flips are considered distinct, there are A244497(n-3) perimeter-magic pentagons. - R. J. Mathar, Mar 10 2025

			for n = 14, a(14) = 1           5
                              6    7
                            3         2
                             10      8
                              1  9  4

R. J. Mathar, Generating perimeter-magic polygons, C++ (2025)

Cf. A380962 (perimeter-magic squares), A380853 (perimeter-magic triangles), A380964 (perimeter-magic hexagons).

A382455 Order 3 perimeter magic squares of magic sum n, all elements distinct and 1 in the set; bracelet symmetry.

Original entry on oeis.org

3, 9, 23, 45, 75, 109, 178, 220, 324, 403, 545, 623, 872, 945, 1238, 1397, 1725, 1878, 2390, 2530, 3087, 3317, 3968, 4212, 5057, 5256, 6186, 6569, 7569, 7893, 9201, 9511, 10890, 11359, 12863, 13340, 15135, 15543, 17492, 18145, 20170, 20739, 23212, 23784, 26325, 27100, 29813, 30598, 33727

Offset: 12

R. J. Mathar, Mar 26 2025

nonn

The order-3 perimeter-magic square has 3 positive integers (=order) per side (4 at corners, 4 within edges, total 8) where the sum of the 3 integers on each side is the same. Here we require that all 8 integers are distinct, that 1 is one of them, and that rotations and flips are not counted separately (recognizing the bracelet symmetry).

R. J. Mathar, Generating perimeter-magic polygons, C++ (2025).

Cf. A380962 (1 not necessarily included), A084569 (elements not necess. distinct).

a(n) = A380962(n)-A380962(n-3).

cp's OEIS Frontend

A380964 Perimeter-magic hexagons of order 3 with magic sum n.

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A380963 The number of perimeter-magic pentagons of order 3 with magic sum n.

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A382455 Order 3 perimeter magic squares of magic sum n, all elements distinct and 1 in the set; bracelet symmetry.

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