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

Author

Derek Holton and Alex Holton, Feb 09 2025

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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)

Formula

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