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A068722 Number of solenoidal flows (flow in = flow out) in a 3 X 3 square array with integer velocities -n .. n. 36

%I

%S 1,35,247,925,2501,5551,10795,19097,31465,49051,73151,105205,146797,

%T 199655,265651,346801,445265,563347,703495,868301,1060501,1282975,

%U 1538747,1830985,2163001,2538251,2960335,3432997,3960125,4545751

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

%C 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

%H Reinhard Zumkeller, <a href="/A068722/b068722.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

%F G.f.: (1+30*x+82*x^2+30*x^3+x^4)/(1-x)^5. - _Colin Barker_, Jul 30 2012

%e Sample flows (. represents a space):

%e Numbers in long rows are on cell walls showing velocity rightward.

%e Numbers in long columns are on cell floors showing velocity downwards.

%e 3 X 3 cell centers are at the intersection of long rows and long columns.

%e n=1:

%e .. 0 . 0 . 0

%e .0. -1. -1 . 0

%e .. 1 . 0. -1

%e .0 . 0 . 0 . 0

%e .. 1 . 0. -1

%e .0 . 1 . 1 . 0

%e .. 0 . 0 . 0

%e n=2:

%e .. 0 . 0 . 0

%e .0. -2. -1 . 0

%e .. 2. -1. -1

%e .0 . 0. -1 . 0

%e .. 2 . 0. -2

%e .0 . 2 . 2 . 0

%e .. 0 . 0 . 0

%o (Haskell)

%o a068722 n = (1 + 2 * n + 2 * n ^ 2) * (1 + 3 * n + 3 * n ^ 2)

%o -- _Reinhard Zumkeller_, Mar 23 2015

%o (PARI) a(n)=(1+2*n+2*n^2)*(1+3*n+3*n^2) \\ _Charles R Greathouse IV_, Apr 08 2016

%Y 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.

%Y Cf. A016754, A255437.

%K nonn,easy

%O 0,2

%A _R. H. Hardin_, Feb 26 2002

%E Formula corrected by _Colin Barker_, Jul 30 2012

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Last modified March 28 14:24 EDT 2020. Contains 333089 sequences. (Running on oeis4.)