A068723 Number of solenoidal flows (flow in = flow out) in a 4 X 4 square array with integer velocities -n .. n.

1, 2053, 160403, 3096791, 28905785, 173429597, 773568107, 2789847759, 8576204625, 23279942261, 57201104003, 129540052071, 274074872265, 547436323149, 1040787422427, 1895867410719, 3326525754145, 5647051042021, 9308792104563, 14946774417783, 23438233878745, 35975222324029, 54153684727627, 80081666851567, 116509584233329

Offset: 0

Author

R. H. Hardin, Feb 26 2002

Keywords

• nonn

Examples

			Sample flows (. represents a space):
Numbers in long rows are on cell walls showing velocity rightward.
Numbers in long columns are on cell floors showing velocity downwards.
4 X 4 cell centers are at the intersection of long rows and long columns.
n=1:
.. 0 . 0 . 0 . 0
.0. -1. -1. -1 . 0
.. 1 . 0 . 0. -1
.0 . 0. -1 . 0 . 0
.. 1 . 1. -1. -1
.0 . 0 . 1 . 0 . 0
.. 1 . 0 . 0. -1
.0 . 1 . 1 . 1 . 0
.. 0 . 0 . 0 . 0
n=2:
.. 0 . 0 . 0 . 0
.0. -2. -1. -2 . 0
.. 2. -1 . 1. -2
.0 . 0. -2 . 0 . 0
.. 2 . 1. -1. -2
.0 . 0 . 1 . 0 . 0
.. 2 . 0 . 0. -2
.0 . 2 . 2 . 2 . 0
.. 0 . 0 . 0 . 0
		

Links

• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A005408 (2 X 2), A068722 (3 X 3), A068724 (5 X 5), A068725 (6 X 6).

By velocity limit 1..13 A068726-A068738.

Formula

Empirical G.f.: (x+1) * (x^8 + 2042*x^7 + 137876*x^6 + 1447150*x^5 + 3462710*x^4 + 1447150*x^3 + 137876*x^2 + 2042*x + 1) / (x-1)^10. [Colin Barker, Jul 31 2012]

Extensions

a(14)-a(24) from Sean A. Irvine, Mar 06 2024