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A222715 The number of binary pattern classes in the (2,n)-rectangular grid with 5 '1's and (2n-5) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

%I A222715 #38 Sep 08 2022 08:46:04
%S A222715 2,14,66,198,508,1092,2156,3876,6606,10626,16478,24570,35672,50344,
%T A222715 69624,94248,125562,164502,212762,271502,342804,428076,529828,649740,
%U A222715 790790,954954,1145718,1365378,1617968,1906128,2234480,2606032,3026034,3497886,4027506
%N A222715 The number of binary pattern classes in the (2,n)-rectangular grid with 5 '1's and (2n-5) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
%H A222715 Vincenzo Librandi, <a href="/A222715/b222715.txt">Table of n, a(n) for n = 3..1000</a>
%H A222715 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-8,6,6,-8,0,3,-1).
%F A222715 a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) -4*(2*n^2-22*n+63)*(-1)^n, with n>8, a(3)=2, a(4)=14, a(5)=66, a(6)=198, a(7)=508, a(8)=1092.
%F A222715 From _Bruno Berselli_, May 29 2013: (Start)
%F A222715 G.f.: 2*x^3*(1+4*x+12*x^2+8*x^3+7*x^4)/((1+x)^3*(1-x)^6).
%F A222715 a(n) = 3*a(n-1) -8*a(n-3) +6*a(n-4) +6*a(n-5) -8*a(n-6) +3*a(n-8) -a(n-9), with n>11.
%F A222715 a(n) = (n-2)*(n-1)*(8*n^3-16*n^2+6*n-15*(-1)^n+15)/120. (End)
%F A222715 a(n) = (1/4)*(binomial(2*n,5) + 2*binomial(n-1,2)*(1/2)*(1-(-1)^n)). [_Yosu Yurramendi_ and María Merino, Aug 21 2013]
%t A222715 Table[(n - 2) (n - 1) ((8 n^3 - 16 n^2 + 6 n - 15 (-1)^n + 15)/120), {n, 3, 40}] (* _Bruno Berselli_, May 30 2013 *)
%t A222715 LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1}, {2, 14, 66, 198, 508, 1092, 2156, 3876, 6606}, 50] (* _T. D. Noe_, Jun 14 2013 *)
%t A222715 CoefficientList[Series[2 (1 + 4 x + 12 x^2 + 8 x^3 + 7 x^4) / ((1 + x)^3 (1 - x)^6), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%o A222715 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x^3*(1+4*x+12*x^2+8*x^3+7*x^4)/((1+x)^3*(1-x)^6)));
%o A222715 (R) a <- vector()
%o A222715     for(n in 1:40) a[n] <- (1/4)*(choose(2*(n+2),5) + 2*choose(n+1,2)*(1/2)*(1-(-1)^n))
%o A222715     a  [_Yosu Yurramendi_ and María Merino, Aug 21 2013]
%o A222715 (Magma) [(1/4)*(Binomial(2*n,5) + 2*Binomial(n-1,2)*(1/2)*(1-(-1)^n)): n in [3..40]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A222715 Cf. A226048.
%K A222715 nonn,easy
%O A222715 3,1
%A A222715 _Yosu Yurramendi_, May 29 2013