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.

2, 14, 66, 198, 508, 1092, 2156, 3876, 6606, 10626, 16478, 24570, 35672, 50344, 69624, 94248, 125562, 164502, 212762, 271502, 342804, 428076, 529828, 649740, 790790, 954954, 1145718, 1365378, 1617968, 1906128, 2234480, 2606032, 3026034, 3497886, 4027506

Offset: 3

Author

Yosu Yurramendi, May 29 2013

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,0,3,-1).

Crossrefs

Cf. A226048.

Programs

Magma

m:=30; R:=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)));

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

Mathematica

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 *)
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 *)
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 *)

R

a <- vector()
    for(n in 1:40) a[n] <- (1/4)*(choose(2*(n+2),5) + 2*choose(n+1,2)*(1/2)*(1-(-1)^n))
    a  [Yosu Yurramendi and María Merino, Aug 21 2013]

Formula

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.

From Bruno Berselli, May 29 2013: (Start)

G.f.: 2*x^3*(1+4*x+12*x^2+8*x^3+7*x^4)/((1+x)^3*(1-x)^6).

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.

a(n) = (n-2)*(n-1)*(8*n^3-16*n^2+6*n-15*(-1)^n+15)/120. (End)

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]