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

3,1

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).

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]

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 *)

Prog

(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)));
(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]
(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

Crossrefs

Cf. A226048.

Sequence in context: A266590 A196977 A254197 * A197162 A109869 A197777

Adjacent sequences: A222712 A222713 A222714 * A222716 A222717 A222718

Keyword

nonn,easy

Author

Yosu Yurramendi, May 29 2013

Status

approved