A108577 Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having all entries < n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 8, 12, 16, 23, 30, 40, 50, 63, 76, 93, 110, 132, 154, 180, 206, 238, 270, 308, 346, 390, 434, 485, 536, 595, 654, 720, 786, 861, 936, 1020, 1104, 1197, 1290, 1393, 1496, 1610, 1724, 1848, 1972, 2108, 2244, 2392, 2540, 2700, 2860

Offset: 1

Author

Thomas Zaslavsky, Jun 11 2005

Keywords

• nonn
• easy

Comments

A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal. The symmetries are those of the square. - Thomas Zaslavsky, Mar 12 2010

Examples

			a(10) = 1 because there is only one symmetry type of 3 X 3 magic square with entries 1,...,9.
		

Links

• Thomas Zaslavsky, Table of n, a(n) for n = 1..10000.
• Matthias Beck and Thomas Zaslavsky, Auxiliary files for "Six little squares".
• Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,2,-2,1,0,-1,2,-1).

Crossrefs

Cf. A108576, A108578, A108579.

Programs

Mathematica

LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 1, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5}, 58] (* Mike Sheppard, Feb 04 2025 *)

Formula

G.f.: (x^10*(2*x^2+1)) / ((1-x^6)*(1-x^4)*(1-x)^2).

a(n) is given by a quasipolynomial of period 12.