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.

0, 0, 1, 6, 14, 32, 55, 94, 140, 208, 285, 390, 506, 656, 819, 1022, 1240, 1504, 1785, 2118, 2470, 2880, 3311, 3806, 4324, 4912, 5525, 6214, 6930, 7728, 8555, 9470, 10416, 11456, 12529, 13702, 14910, 16224, 17575, 19038, 20540, 22160, 23821, 25606, 27434, 29392

Offset

0,4

Comments

Also the edge count of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Black Bishop Graph

Eric Weisstein's World of Mathematics, Edge Count

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Formula

a(n) = A000330(n) + A142150(n) = (n-1)*(4*n^2 - 2*n - 3*(-1)^n + 3)/12.

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.

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.

G.f.: x^2*(1 + 4*x + x^2 + 2*x^3)/((1+x)^2*(1-x)^4). - Bruno Berselli, May 29 2013

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

a(n) = A005993(n-2) + A199771(n-1), n >= 2. - Christopher Hunt Gribble, Mar 02 2014

Maple

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

Mathematica

Table[(n - 1)*(4*n^2 - 2*n - 3*(-1)^n + 3)/12, {n, 0, 40}] (* Bruno Berselli, May 29 2013 *)
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 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 6, 14, 32, 55}, 20] (* Eric W. Weisstein, Jun 27 2017 *)

Prog

(R) a <- vector()
    for(n in 0:40) a[n] <- (1/4)*(choose(2*(n-1), 3) + 2*choose(n-2, 1)*(1/2)*(1+(-1)^n))
    a  # Yosu Yurramendi and María Merino, Aug 21 2013
(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

Crossrefs

Cf. A226048, A000330.

Cf. A289179 (edge count of white bishop graph).

Sequence in context: A273365 A271996 A199705 * A332724 A078836 A340735

Adjacent sequences: A225969 A225970 A225971 * A225973 A225974 A225975

Keyword

nonn,easy

Author

Yosu Yurramendi, May 26 2013

Extensions

More terms from Vincenzo Librandi, Sep 04 2013

Status

approved