A202094 Number of (n+2) X 4 binary arrays avoiding patterns 001 and 011 in rows and columns.

324, 1296, 3600, 10000, 22500, 50625, 99225, 194481, 345744, 614656, 1016064, 1679616, 2624400, 4100625, 6125625, 9150625, 13176900, 18974736, 26501904, 37015056, 50381604, 68574961, 91298025, 121550625, 158760000, 207360000, 266342400, 342102016, 432972864, 547981281, 684502569, 855036081

Offset

1,1

Comments

Part of a family of width-w binary arrays avoiding 001 and 011 (w=3..9: A202093-A202099) with common formula a(n) = C(alpha+E,E)*C(alpha+O,O)*C(beta+E,E)*C(beta+O,O) where E=ceil(w/2), O=floor(w/2), alpha=floor((n+3)/2), beta=floor((n+2)/2). Since w=4 here, a(n) is always a perfect square. - Christian Krause, Jun 27 2026

Links

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

Index entries for linear recurrences with constant coefficients, signature (2,6,-14,-14,42,14,-70,0,70,-14,-42,14,14,-6,-2,1).

Formula

a(n) = 2*a(n-1) +6*a(n-2) -14*a(n-3) -14*a(n-4) +42*a(n-5) +14*a(n-6) -70*a(n-7) +70*a(n-9) -14*a(n-10) -42*a(n-11) +14*a(n-12) +14*a(n-13) -6*a(n-14) -2*a(n-15) +a(n-16). - Proved by Christian Krause, Jun 27 2026

From Christian Krause, Jun 27 2026: (Start)

a(n) = (n+4)^4 * (n+6)^4 / 4096 for n even.

a(n) = (n+3)^2 * (n+5)^4 * (n+7)^2 / 4096 for n odd. (End)

From Amiram Eldar, Jun 28 2026: (Start)

Sum_{n>=1} 1/a(n) = 8*Pi^4/15 + 60*Pi^2 - 52174/81.

Sum_{n>=1} (-1)^(n+1)/a(n) = 38710/81 - 8*Pi^4/45 - 140*Pi^2/3. (End)

Example

Some solutions for n=7
..1..1..1..1....1..1..1..1....1..1..0..1....1..1..0..1....1..1..1..1
..1..1..1..1....1..1..0..1....1..1..1..0....1..1..0..1....1..1..1..0
..0..1..0..1....1..1..1..1....1..1..0..0....0..1..0..0....0..1..0..1
..1..1..1..0....0..1..0..1....1..1..1..0....1..1..0..0....1..1..0..0
..0..1..0..0....1..1..0..1....1..1..0..0....0..0..0..0....0..1..0..0
..0..0..0..0....0..1..0..1....1..1..0..0....1..1..0..0....1..1..0..0
..0..1..0..0....1..1..0..0....1..1..0..0....0..0..0..0....0..1..0..0
..0..0..0..0....0..0..0..0....0..1..0..0....0..1..0..0....0..1..0..0
..0..1..0..0....0..1..0..0....0..1..0..0....0..0..0..0....0..0..0..0

Mathematica

a[n_] := If[EvenQ[n], (n+4)^4 * (n+6)^4, (n+3)^2 * (n+5)^4 * (n+7)^2] / 4096; Array[a, 32] (* Amiram Eldar, Jun 28 2026 *)

Crossrefs

Column 2 of A202100.

Cf. A202093.

Sequence in context: A014792 A287634 A329613 * A202486 A305070 A229538

Adjacent sequences: A202091 A202092 A202093 * A202095 A202096 A202097

Keyword

nonn,easy

Author

R. H. Hardin, Dec 11 2011

Status

approved