A122751 Number of essentially different semi-magic squares of order 3 with semimagic sum n.

1, 2, 7, 14, 29, 49, 83, 127, 192, 273, 384, 519, 694, 902, 1162, 1466, 1835, 2260, 2765, 3340, 4011, 4767, 5637, 6609, 7714, 8939, 10318, 11837, 13532, 15388, 17444, 19684, 22149, 24822, 27747, 30906, 34345, 38045, 42055, 46355, 50996, 55957, 61292

Offset: 3

Author

Christoph Gerber (christoph.gerber(AT)phbern.ch), Jun 25 2007

Keywords

• nonn

Examples

			a(4)=2 because there are 2 essentially different semi-magic squares of order 3 with semi-magic sum 4: [1,1,2; 1,2,1; 2,1,1] and [1,1,2; 2,1,1; 1,2,1].
		

References

• Christoph Gerber, "Zum Abzahlen semimagischer Quadrate" [Apparently unpublished. - R. J. Mathar, Nov 13 2011]
• P. A. MacMahon, Combinatory Analysis, Vol II; Chelsea, New York, 1960.

Links

• Harvey P. Dale, Table of n, a(n) for n = 3..1000
• Christoph Gerber, More information [?Broken link]
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,4,-4,2,2,-3,1).

Programs

Maple

A131292:=proc(n) local d,e: if (n mod 4) in {0,2} then d:=-1/8 fi: if (n mod 4) in {1,3} then d:=3/32 fi: if (n mod 4) in {0} then e:=0 fi: if (n mod 4) in {1} then e:=-7/64 fi: if (n mod 4) in {2} then e:=1/8 fi: if (n mod 4) in {3} then e:=1/64 fi: return 1/64*n^4-1/32*n^3+1/32*n^2+d*n+e: end proc:

Mathematica

LinearRecurrence[{3,-2,-2,4,-4,2,2,-3,1},{1,2,7,14,29,49,83,127,192},50] (* Harvey P. Dale, Jan 26 2017 *)

Formula

a(n) = 1/64*n^4-1/32*n^3+1/32*n^2+d*n+e with: d:=-1/8 if n=0 or n=2 (mod 4) d:=3/32 if n=1 or n=3 (mod 4) e:=0 if n=0 (mod 4) e:=-7/64 if n=1 (mod 4) e:=1/8 if n=2 (mod 4) e:=1/64 if n=3 (mod 4).

G.f.: -x^3*(1-x+3*x^2-x^3+x^4) / ( (1+x^2)*(1+x)^2*(x-1)^5 ). - R. J. Mathar, Nov 13 2011

a(n) = (2*n*(n-1)*(n^2-n+1)-7*(2*n-1)*(-1)^n-8*(-1)^((2*n-1+(-1)^n)/4)+1)/128. - Luce ETIENNE, Oct 29 2017