A202195 Number of (n+2) X 3 binary arrays avoiding patterns 001 and 101 in rows and columns.

108, 240, 450, 756, 1176, 1728, 2430, 3300, 4356, 5616, 7098, 8820, 10800, 13056, 15606, 18468, 21660, 25200, 29106, 33396, 38088, 43200, 48750, 54756, 61236, 68208, 75690, 83700, 92256, 101376, 111078, 121380, 132300, 143856, 156066, 168948, 182520, 196800, 211806, 227556, 244068, 261360, 279450, 298356

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

Christian Krause, Proof of formula, Jun 24 2026

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = 3*(n+3)*(n+2)^2 = 3*A011379(n+2). [proved by Christian Krause, Jun 24 2026]

From Colin Barker, Mar 03 2018: (Start)

G.f.: 6*x*(18 - 32*x + 23*x^2 - 6*x^3) / (1 - x)^4.

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

From Amiram Eldar, Jun 16 2026: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/18 - 19/36.

Sum_{n>=1} (-1)^(n+1)/a(n) = 7/36 + Pi^2/36 - 2*log(2)/3. (End)

Example

Some solutions for n=10:
  0 0 0    0 0 0    1 0 0    1 0 0    1 0 0    0 1 1    0 1 1
  1 1 1    0 1 1    0 1 1    1 1 0    1 1 1    1 1 1    1 1 1
  1 1 0    0 1 1    0 1 1    0 1 0    1 1 1    1 1 1    1 1 1
  1 1 0    0 1 1    0 1 1    0 1 0    1 1 0    1 1 1    1 1 1
  1 1 0    0 1 1    0 1 0    0 1 0    1 0 0    1 1 1    1 1 1
  0 1 0    0 1 1    0 0 0    0 1 0    0 0 0    1 1 1    1 1 0
  0 1 0    0 1 1    0 0 0    0 1 0    0 0 0    1 1 0    1 0 0
  0 1 0    0 1 1    0 0 0    0 0 0    0 0 0    1 1 0    0 0 0
  0 1 0    0 1 0    0 0 0    0 0 0    0 0 0    0 1 0    0 0 0
  0 1 0    0 1 0    0 0 0    0 0 0    0 0 0    0 1 0    0 0 0
  0 1 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0
  0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0

Mathematica

A202195[n_] := 3*(n+3)*(n+2)^2; Array[A202195, 50] (* Paolo Xausa, Jun 25 2026 *)

Crossrefs

Column 1 of A202202.

Sequence in context: A275996 A235292 A202202 * A242867 A396459 A255091

Adjacent sequences: A202192 A202193 A202194 * A202196 A202197 A202198

Keyword

nonn,easy

Author

R. H. Hardin, Dec 14 2011

Status

approved