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

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).

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

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.

Original entry on oeis.org

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

A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 13, 14, 25, 37, 47, 58, 89, 98, 126, 159, 188, 219, 276, 303, 362, 423, 478, 536, 633, 688, 781, 881, 973, 1068, 1211, 1301, 1443, 1589, 1724, 1866, 2066, 2202, 2396, 2598, 2790, 2986, 3250, 3439, 3699, 3967, 4219, 4480, 4819, 5071

Offset: 1

Derek Holton and Alex Holton, Feb 06 2025

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

If the numbers do not need to be distinct and rotations and reflections are counted separately we get A019298(n-2). If the numbers do not need to be distinct but rotations and reflections do not count separately we get A006918(n-2). If the six numbers must be distinct and reflections and rotations count separately we get 6*a(n). - R. J. Mathar, Feb 27 2025

			The a(9) = 1 solution is:
       1
     5   6
   3   4   2

R. J. Mathar, Illustrations - Examples (2025)
Index entries for linear recurrences with constant coefficients, signature (-1,0,2,3,2,0,-3,-4,-3,0,2,3,2,0,-1,-1).

Cf. A342467, A380105.

A380853 := proc(n)
    -3412+2353*n+30*n^3-480*n^2+(135*n-900)*(-1)^n ;
    %-144*(2*A010891(n)+A010891(n-1)+2*A010891(n-2)) ;
    %-160*(17*A049347(n)+8*A049347(n-1)) ;
    %-360*A057077(n) ;
    %+480*(-1)^n*(A099254(n)-A099254(n-1)) ;
    %/720 ;
end proc:
seq(A380853(n),n=1..40) ; # R. J. Mathar, Feb 27 2025

a(n) = my(c=0, t); for(x=3, n-5, t=n-x; for(y=2, min(x-1, t-1), for(z=1, y-1, if(#Set([x, y, z, t-y, t-z, n-y-z])==6, c++)))); c; \\ Jinyuan Wang, Feb 07 2025

G.f.: x^9*(1 + 4*x + 8*x^2 + 16*x^3 + 18*x^4 + 18*x^5 + 15*x^6 + 10*x^7)/((1 - x)^4*(1 + 2*x + 2*x^2 + x^3)^2*(1 + x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6)). - Stefano Spezia, Feb 08 2025

A380105(n) = a(n)-a(n-3). - R. J. Mathar, Mar 13 2025

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User: Derek Holton

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

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

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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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A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

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