A068744 Number of potential flows in 3 X 3 array with integer velocities in -n..n, i.e., number of 3 X 3 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

1, 1665, 87825, 1253329, 9230193, 45642289, 172989921, 542131425, 1473095713, 3582226465, 7970825457, 16492629297, 32119620625, 59427841617, 105227044417, 179360179905, 295700892993, 473379359425, 738268965841

Offset: 0

Author

R. H. Hardin, Feb 27 2002

Keywords

• nonn

Comments

Let y = 2*n - 1; Then apparently a(n) = y^2*(529*y^6 + 910*y^4 + 721*y^2 + 360)/2520. See A068745 (4 X 4) and A063496 (2 X 2), which is y*(2*y^2 + 1)/3 under the same transformation. Suggests total degree N X N-1, with a factor y or y^2 to make the remaining polynomial even. - R. H. Hardin, Jan 02 2007

Crossrefs

2 X 2 A063496, 4 X 4 A068745, 5 X 5 A068746, 6 X 6 A068747, by velocity limit 1..14 A068748-A068761, solenoidal flows A068722-A068738.

Formula

Empirical G.f.: -(x^8+1656*x^7 +72876*x^6 +522760*x^5 +972198*x^4 +522760*x^3 +72876*x^2 +1656*x +1)/(x-1)^9. [Colin Barker, Jul 31 2012]

Empirical: 315*a(n) = (4232*n^6 +12696*n^5 +17690*n^4 +14220*n^3 +7058*n^2 +2064*n +315) *(1+2*n)^2. - R. J. Mathar, Nov 09 2018