A202198 Number of (n+2) X 6 binary arrays avoiding patterns 001 and 101 in rows and columns.

756, 2688, 7560, 18144, 38808, 76032, 138996, 240240, 396396, 628992, 965328, 1439424, 2093040, 2976768, 4151196, 5688144, 7671972, 10200960, 13388760, 17365920, 22281480, 28304640, 35626500, 44461872, 55051164, 67662336, 82592928, 100172160, 120763104, 144764928, 172615212, 204792336, 241817940

Offset

1,1

Comments

Part of the family a(n) = 2*w*(n+2)*C(n+w,w-1) for width-w binary arrays avoiding patterns 001 and 101 (A202195-A202201 for w=3..9). - Christian Krause, Jun 24 2026

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (n+6)*(n+5)*(n+4)*(n+3)*(n+2)^2/10. [proved by Christian Krause, Jun 24 2026]

From Colin Barker, May 27 2018: (Start)

G.f.: 12*x*(63 - 217*x + 385*x^2 - 399*x^3 + 245*x^4 - 83*x^5 + 12*x^6) / (1 - x)^7.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7. (End)

From Amiram Eldar, Jun 28 2026: (Start)

Sum_{n>=1} 1/a(n) = 5*Pi^2/72 - 1181/1728.

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi^2/144 - 40*log(2)/9 + 4733/1728. (End)

Example

Some solutions for n = 3:
  0  1  1  1  1  1    1  1  1  0  0  0    0  1  1  0  0  0    1  1  1  1  1  1
  1  1  1  1  1  1    1  1  1  0  0  0    0  1  1  0  0  0    1  1  1  1  1  1
  0  1  1  1  0  0    1  1  0  0  0  0    0  1  1  0  0  0    1  1  1  1  1  1
  0  1  1  0  0  0    1  1  0  0  0  0    0  1  1  0  0  0    1  1  1  0  0  0
  0  1  1  0  0  0    1  1  0  0  0  0    0  1  1  0  0  0    0  1  1  0  0  0

Mathematica

A202198[n_] := (n+6)*(n+5)*(n+4)*(n+3)*(n+2)^2/10;
Array[A202198, 35] (* Paolo Xausa, Jun 25 2026 *)

Crossrefs

Column 4 of A202202.

Cf. A202195.

Sequence in context: A252015 A108374 A252007 * A158657 A004010 A320686

Adjacent sequences: A202195 A202196 A202197 * A202199 A202200 A202201

Keyword

nonn,easy

Author

R. H. Hardin, Dec 14 2011

Status

approved