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A225972 The number of binary pattern classes in the (2,n)-rectangular grid with 3 '1's and (2n-3) '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 A225972 #64 Feb 16 2025 08:33:19
%S A225972 0,0,1,6,14,32,55,94,140,208,285,390,506,656,819,1022,1240,1504,1785,
%T A225972 2118,2470,2880,3311,3806,4324,4912,5525,6214,6930,7728,8555,9470,
%U A225972 10416,11456,12529,13702,14910,16224,17575,19038,20540,22160,23821,25606,27434,29392
%N A225972 The number of binary pattern classes in the (2,n)-rectangular grid with 3 '1's and (2n-3) '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.
%C A225972 Also the edge count of the n X n black bishop graph. - _Eric W. Weisstein_, Jun 26 2017
%H A225972 Vincenzo Librandi, <a href="/A225972/b225972.txt">Table of n, a(n) for n = 0..1000</a>
%H A225972 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>
%H A225972 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>
%H A225972 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A225972 a(n) = A000330(n) + A142150(n) = (n-1)*(4*n^2 - 2*n - 3*(-1)^n + 3)/12.
%F A225972 a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) with n > 5, a(0)=0, a(1)=0, a(2)=1, a(3)=6, a(4)=14, a(5)=32.
%F A225972 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 4*(n-4)*(-1)^n with n > 3, a(0)=0, a(1)=0, a(2)=1, a(3)=6.
%F A225972 G.f.: x^2*(1 + 4*x + x^2 + 2*x^3)/((1+x)^2*(1-x)^4). - _Bruno Berselli_, May 29 2013
%F A225972 a(n) = (1/4)*(binomial(2*(n-1),3) + 2*binomial(n-2,1)*(1/2)*(1+(-1)^n)). - _Yosu Yurramendi_ and _María Merino_, Aug 21 2013
%F A225972 a(n) = A005993(n-2) + A199771(n-1), n >= 2. - _Christopher Hunt Gribble_, Mar 02 2014
%p A225972 A225972:=n->(n-1)*(4*n^2-2*n-3*(-1)^n+3)/12; seq(A225972(n), n=0..40); # _Wesley Ivan Hurt_, Mar 02 2014
%t A225972 Table[(n - 1)*(4*n^2 - 2*n - 3*(-1)^n + 3)/12, {n, 0, 40}] (* _Bruno Berselli_, May 29 2013 *)
%t A225972 CoefficientList[Series[x^2 (1 + 4 x + x^2 + 2 x^3) / ((1 + x)^2 (1 - x)^4), {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%t A225972 LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 6, 14, 32, 55}, 20] (* _Eric W. Weisstein_, Jun 27 2017 *)
%o A225972 (R) a <- vector()
%o A225972     for(n in 0:40) a[n] <- (1/4)*(choose(2*(n-1),3) + 2*choose(n-2,1)*(1/2)*(1+(-1)^n))
%o A225972     a  # _Yosu Yurramendi_ and _María Merino_, Aug 21 2013
%o A225972 (Magma) [(1/4)*(Binomial(2*(n-1),3)+2*Binomial(n-2,1)*(1/2)*(1+(-1)^n)): n in [1..50]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A225972 Cf. A226048, A000330.
%Y A225972 Cf. A289179 (edge count of white bishop graph).
%K A225972 nonn,easy
%O A225972 0,4
%A A225972 _Yosu Yurramendi_, May 26 2013
%E A225972 More terms from _Vincenzo Librandi_, Sep 04 2013