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
Keywords
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- Christoph Gerber, "Zum Abzahlen semimagischer Quadrate" [Apparently unpublished.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