A380853 -id:A380853 - cp's OEIS frontend

A380962 Number of ways to place eight distinct positive integers on a square, four on the corners and four on the sides such that the sum of the three values on each side is n.

3, 9, 23, 48, 84, 132, 226, 304, 456, 629, 849, 1079, 1501, 1794, 2317, 2898, 3519, 4195, 5288, 6049, 7282, 8605, 10017, 11494, 13662, 15273, 17680, 20231, 22842, 25573, 29432, 32353, 36463, 40791, 45216, 49803, 55926, 60759, 67295, 74071, 80929, 88034, 97283, 104713, 114359, 124383, 134526, 144957, 158110

Offset: 12

Derek Holton and Alex Holton, Feb 09 2025

Solutions differing by only rotation or reflections are not counted separately.

			for n = 12, one of the a(12) = 3 possible arrangements of numbers is
  2  6  4
  9     5
  1  8  3
The 3 numbers of each side sum to 12, eg. 2+9+1 = 12.

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

Cf. A380853 (order 3 perimeter magic triangles), A005994 (8 elements need not be distinct), A006325 (8 elements need not be distinct, rotat+flips count separately)

Conjecture: a(n)= -2*a(n-1) -3*a(n-2) -2*a(n-3) +3*a(n-5) +6*a(n-6) +8*a(n-7) +9*a(n-8) +7*a(n-9) +3*a(n-10) -4*a(n-11) -10*a(n-12) -15*a(n-13) -16*a(n-14) -14*a(n-15) -8*a(n-16) +8*a(n-18) +14*a(n-19) +16*a(n-20) +15*a(n-21) +10*a(n-22) +4*a(n-23) -3*a(n-24) -7*a(n-25) -9*a(n-26) -8*a(n-27) -6*a(n-28) -3*a(n-29) +2*a(n-31) +3*a(n-32) +2*a(n-33) +a(n-34). - R. J. Mathar, Mar 04 2025

Conjecture: g.f. ( -x^12 *(3045*x^12 +2826*x^11 +2520*x^10 +2079*x^9 +1625*x^8 +1173*x^7 +793*x^6 +267*x^4 +481*x^5 +98*x^22 +236*x^21 +491*x^20 +796*x^19 +1231*x^18 +1673*x^17 +2187*x^16 +2580*x^15 +2906*x^14 +3038*x^13 +127*x^3 +3 +15*x +50*x^2) ) / ( (x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+1)^2 *(1+x)^3 *(1+x+x^2)^3 *(x-1)^5 ). - R. J. Mathar, Mar 04 2025

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

Original entry on oeis.org

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

A380105 Perimeter-magic triangles of order 3 with magic sum n, bracelet symmetry, and minimum term 1.

Original entry on oeis.org

1, 3, 5, 12, 11, 20, 24, 33, 33, 52, 51, 68, 70, 90, 93, 117, 115, 143, 147, 175, 174, 210, 210, 245, 248, 285, 287, 330, 328, 375, 378, 423, 423, 477, 478, 530, 532, 588, 590, 652, 649, 713, 717, 780, 781, 852, 852, 923, 925, 1000, 1001, 1080, 1078, 1160, 1165, 1245, 1245, 1335, 1335

Offset: 9

R. J. Mathar, Mar 11 2025

nonn

The setup is similar to A380853: triangles with 3 positive integers per side, all 6 values distinct, each side's sum equal to n, respecting the bracelet symmetry (rotations and flips generate equivalent triangles). But here 1 must be present at least once among the 6 values. The formula reflects that one can generate magic triangles where 1 is no longer present by adding 1 to each of the 6 integers, which increases the magic sum by 3.

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

Cf. A380853 (perimeter-magic triangles).

a(n) = A380853(n)-A380853(n-3).

G.f.: A380853(x)*(1-x^3).

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).

A380980 Place 2n distinct positive integers on an n-gon, n on the vertices and n on the sides such that the sums of the three values on all sides are equal. a(n) is the minimal sum of all the integers used.

Original entry on oeis.org

21, 38, 55, 81, 105, 140

Offset: 3

Ivan N. Ianakiev, Feb 10 2025

			For n=7 the first 14 positive integers suffice. Their permutation is 1,11,7,8,4,9,6,10,3,14,2,12,5,13 and the sum on each side of the heptagon is 19.

Cf. A380853.

mod[n_]:=Append[n,First[n]];
plus[n_]:=Table[mod[n][[i]]+mod[n][[i+1]],{i,1,Length[mod[n]]-1}];
goodPermutations[n_]:=Select[Permutations[n],Length[plus[#]]==Length[Union[plus[#]]]&];
min[n_]:=Min[Max/@plus/@goodPermutations[Range[n]]];
bestPermutation[n_]:=Select[goodPermutations[Range[n]],Max[plus[#]]==min[n]&,1];
plusBP[n_]:=plus/@bestPermutation[n]; max[n_]:=Max[Max/@plusBP[n]];
unit[n_]:=max[n]+n+1; sum[n_]:=n*unit[n]-Total@@plusBP[n]+n*(n+1)/2; sum/@Range[3,8]

cp's OEIS Frontend

A380962 Number of ways to place eight distinct positive integers on a square, four on the corners and four on the sides such that the sum of the three values on each side is n.

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

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A380105 Perimeter-magic triangles of order 3 with magic sum n, bracelet symmetry, and minimum term 1.

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

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A380980 Place 2n distinct positive integers on an n-gon, n on the vertices and n on the sides such that the sums of the three values on all sides are equal. a(n) is the minimal sum of all the integers used.

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Mathematica