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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.

%I A068722 #26 Mar 10 2024 09:33:52
%S A068722 1,35,247,925,2501,5551,10795,19097,31465,49051,73151,105205,146797,
%T A068722 199655,265651,346801,445265,563347,703495,868301,1060501,1282975,
%U A068722 1538747,1830985,2163001,2538251,2960335,3432997,3960125,4545751,5194051,5909345,6696097,7558915
%N A068722 Number of solenoidal flows (flow in = flow out) in a 3 X 3 square array with integer velocities -n .. n.
%C A068722 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 A068722 Reinhard Zumkeller, <a href="/A068722/b068722.txt">Table of n, a(n) for n = 0..10000</a>
%H A068722 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A068722 a(n) = (1+2*n+2*n^2) * (1+3*n+3*n^2).
%F A068722 G.f.: (1+30*x+82*x^2+30*x^3+x^4)/(1-x)^5. - _Colin Barker_, Jul 30 2012
%F A068722 E.g.f.: exp(x)*(1 + 34*x + 89*x^2 + 48*x^3 + 6*x^4). - _Stefano Spezia_, Mar 10 2024
%e A068722 Sample flows (. represents a space):
%e A068722 Numbers in long rows are on cell walls showing velocity rightward.
%e A068722 Numbers in long columns are on cell floors showing velocity downwards.
%e A068722 3 X 3 cell centers are at the intersection of long rows and long columns.
%e A068722 n=1:
%e A068722 .. 0 . 0 . 0
%e A068722 .0. -1. -1 . 0
%e A068722 .. 1 . 0. -1
%e A068722 .0 . 0 . 0 . 0
%e A068722 .. 1 . 0. -1
%e A068722 .0 . 1 . 1 . 0
%e A068722 .. 0 . 0 . 0
%e A068722 n=2:
%e A068722 .. 0 . 0 . 0
%e A068722 .0. -2. -1 . 0
%e A068722 .. 2. -1. -1
%e A068722 .0 . 0. -1 . 0
%e A068722 .. 2 . 0. -2
%e A068722 .0 . 2 . 2 . 0
%e A068722 .. 0 . 0 . 0
%o A068722 (Haskell)
%o A068722 a068722 n = (1 + 2 * n + 2 * n ^ 2) * (1 + 3 * n + 3 * n ^ 2)
%o A068722 -- _Reinhard Zumkeller_, Mar 23 2015
%o A068722 (PARI) a(n)=(1+2*n+2*n^2)*(1+3*n+3*n^2) \\ _Charles R Greathouse IV_, Apr 08 2016
%Y A068722 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 A068722 Cf. A016754, A255437.
%K A068722 nonn,easy
%O A068722 0,2
%A A068722 _R. H. Hardin_, Feb 26 2002
%E A068722 Formula corrected by _Colin Barker_, Jul 30 2012