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

12,1

Comments

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

Links

Table of n, a(n) for n=12..60.

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

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

a(n) = (1260*n^4-37800*n^3+409150*n^2-1910628*n+3020679)/60480 - (n mod 2)*(4*n^2-83*n+384)/16 + (((n+2) mod 3)-(n mod 3))*(7*n^2-162*n+882)/54 + ((20*n-323)*(n mod 4) -73*((n+1) mod 4) -(20*n-361)*((n+2) mod 4) +35*((n+3) mod 4))/128 +((2*n+4) mod 5)*9/5 + (((n+3) mod 6)-((n+2) mod 6))*7/12 + ((5*n+6) mod 7)*2/7 + (((n+2) mod 8) -((n+6) mod 8) +2*((n+5) mod 8) -2*((n+3) mod 8))/8. - Hoang Xuan Thanh, May 30 2026

The conjectures above are true. - Hoang Xuan Thanh, May 30 2026

Example

for n = 12, a(12) = 3: three possible arrangements of numbers is
  3  8  1         2  9  1         1  9  2
  7     5    ,    7     6    ,    8     6
  2  4  6         3  4  5         3  5  4
(Type "alpha")  (Type "C")      (Type "Z")

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)

Sequence in context: A382455 A147271 A230119 * A146693 A227259 A064551

Adjacent sequences: A380959 A380960 A380961 * A380963 A380964 A380965

Keyword

nonn,easy

Author

Derek Holton and Alex Holton, Feb 09 2025

Status

approved