A202201 Number of (n+2) X 9 binary arrays avoiding patterns 001 and 101 in rows and columns.

2430, 11880, 44550, 138996, 378378, 926640, 2084940, 4375800, 8664084, 16325712, 29476980, 51279480, 86337900, 141210432, 225054126, 350430300, 534298050, 799227000, 1174863690, 1699689420, 2423110950, 3407929200, 4733235000, 6497785008, 8823915144, 11862053280, 15795897480, 20848330800

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 25 2026

Links

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

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

a(n) = (n+9)*(n+8)*(n+7)*(n+6)*(n+5)*(n+4)*(n+3)*(n+2)^2/2240. [proved by Christian Krause, Jun 25 2026]

From Colin Barker, May 27 2018: (Start)

G.f.: 18*x*(135 - 690*x + 1950*x^2 - 3528*x^3 + 4326*x^4 - 3660*x^5 + 2115*x^6 - 800*x^7 + 179*x^8 - 18*x^9) / (1 - x)^10.

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 10. (End)

From Amiram Eldar, Jun 28 2026: (Start)

Sum_{n>=1} 1/a(n) = 2*Pi^2/27 - 24163/33075.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/27 + 1370464/99225 - 19328*log(2)/945. (End)

Example

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

Mathematica

a[n_] := (n+9)*(n+8)*(n+7)*(n+6)*(n+5)*(n+4)*(n+3)*(n+2)^2/2240; Array[a, 28] (* Amiram Eldar, Jun 28 2026 *)

Crossrefs

Column 7 of A202202.

Cf. A202195.

Sequence in context: A186874 A345174 A345175 * A147984 A229871 A192767

Adjacent sequences: A202198 A202199 A202200 * A202202 A202203 A202204

Keyword

nonn,easy,changed

Author

R. H. Hardin, Dec 14 2011

Status

approved