A068722 Number of solenoidal flows (flow in = flow out) in a 3 X 3 square array with integer velocities -n .. n.

1, 35, 247, 925, 2501, 5551, 10795, 19097, 31465, 49051, 73151, 105205, 146797, 199655, 265651, 346801, 445265, 563347, 703495, 868301, 1060501, 1282975, 1538747, 1830985, 2163001, 2538251, 2960335, 3432997, 3960125, 4545751, 5194051, 5909345, 6696097, 7558915

Offset: 0

Author

R. H. Hardin, Feb 26 2002

Keywords

• nonn
• easy

Comments

Conjecture: A255437(a(n)) = 2*n+1, i.e. a(n) = gives the position of the first occurrence of 2*n+1 in A255437. - Reinhard Zumkeller, Mar 23 2015

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.
3 X 3 cell centers are at the intersection of long rows and long columns.
n=1:
.. 0 . 0 . 0
.0. -1. -1 . 0
.. 1 . 0. -1
.0 . 0 . 0 . 0
.. 1 . 0. -1
.0 . 1 . 1 . 0
.. 0 . 0 . 0
n=2:
.. 0 . 0 . 0
.0. -2. -1 . 0
.. 2. -1. -1
.0 . 0. -1 . 0
.. 2 . 0. -2
.0 . 2 . 2 . 0
.. 0 . 0 . 0
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. 2 X 2=1, 3, 5, 7..., 4 X 4 A068723, 5 X 5 A068724, 6x6 A068725, by velocity limit 1..13 A068726-A068738.

Cf. A016754, A255437.

Programs

Haskell

a068722 n = (1 + 2 * n + 2 * n ^ 2) * (1 + 3 * n + 3 * n ^ 2)
-- Reinhard Zumkeller, Mar 23 2015

PARI

a(n)=(1+2*n+2*n^2)*(1+3*n+3*n^2) \\ Charles R Greathouse IV, Apr 08 2016

Formula

a(n) = (1+2*n+2*n^2) * (1+3*n+3*n^2).

G.f.: (1+30*x+82*x^2+30*x^3+x^4)/(1-x)^5. - Colin Barker, Jul 30 2012

E.g.f.: exp(x)*(1 + 34*x + 89*x^2 + 48*x^3 + 6*x^4). - Stefano Spezia, Mar 10 2024

Extensions

Formula corrected by Colin Barker, Jul 30 2012