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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.

%I A380962 #20 Mar 19 2025 05:55:48
%S A380962 3,9,23,48,84,132,226,304,456,629,849,1079,1501,1794,2317,2898,3519,
%T A380962 4195,5288,6049,7282,8605,10017,11494,13662,15273,17680,20231,22842,
%U A380962 25573,29432,32353,36463,40791,45216,49803,55926,60759,67295,74071,80929,88034,97283,104713,114359,124383,134526,144957,158110
%N 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.
%C A380962 Solutions differing by only rotation or reflections are not counted separately.
%H A380962 R. J. Mathar, <a href="https://zenodo.org/records/15001365">Generating perimeter-magic polygons</a>, C++ (2025)
%F A380962 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
%F A380962 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
%e A380962 for n = 12, one of the a(12) = 3 possible arrangements of numbers is
%e A380962   2  6  4
%e A380962   9     5
%e A380962   1  8  3
%e A380962 The 3 numbers of each side sum to 12, eg. 2+9+1 = 12.
%Y A380962 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)
%K A380962 nonn,easy
%O A380962 12,1
%A A380962 _Derek Holton_ and Alex Holton, Feb 09 2025