A202093 Number of (n+2) X 3 binary arrays avoiding patterns 001 and 011 in rows and columns.

108, 324, 720, 1600, 3000, 5625, 9450, 15876, 24696, 38416, 56448, 82944, 116640, 164025, 222750, 302500, 399300, 527076, 679536, 876096, 1107288, 1399489, 1739010, 2160900, 2646000, 3240000, 3916800, 4734976, 5659776, 6765201, 8005878, 9474084, 11111580, 13032100, 15162000, 17640000, 20374200

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). - Christian Krause, Jun 26 2026

Links

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

Christian Krause, Proof of formula, Jun 26 2026

Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Formula

a(n) = 2*a(n-1) +4*a(n-2) -10*a(n-3) -5*a(n-4) +20*a(n-5) -20*a(n-7) +5*a(n-8) +10*a(n-9) -4*a(n-10) -2*a(n-11) +a(n-12). [proved by Christian Krause, Jun 26 2026]

From Colin Barker, Feb 20 2018: (Start)

G.f.: x*(108 + 108*x - 360*x^2 - 56*x^3 + 700*x^4 - 115*x^5 - 680*x^6 + 236*x^7 + 334*x^8 - 155*x^9 - 66*x^10 + 36*x^11) / ((1 - x)^7*(1 + x)^5).

a(n) = (n^6 + 28*n^5 + 324*n^4 + 1984*n^3 + 6784*n^2 + 12288*n + 9216) / 256 for n even.

a(n) = (n^6 + 28*n^5 + 321*n^4 + 1928*n^3 + 6395*n^2 + 11100*n + 7875) / 256 for n odd.

(End) [proved by Christian Krause, Jun 26 2026]

From Amiram Eldar, Jun 28 2026: (Start)

Sum_{n>=1} 1/a(n) = 2*Pi^4/45 + 11*Pi^2/3 - 4*zeta(3) - 1285/36.

Sum_{n>=1} (-1)^(n+1)/a(n) = 12*zeta(3) + 229/36 - 2*Pi^4/45 - 5*Pi^2/3. (End)

Example

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

Mathematica

a[n_] := If[EvenQ[n], (n+4)^4 * (n+6)^2, (n+7) * (n+3)^2 * (n+5)^3] / 256; Array[a, 37] (* Amiram Eldar, Jun 28 2026 *)

Crossrefs

Column 1 of A202100.

Sequence in context: A202100 A202492 A369420 * A202485 A202317 A202310

Adjacent sequences: A202090 A202091 A202092 * A202094 A202095 A202096

Keyword

nonn,easy,changed

Author

R. H. Hardin, Dec 11 2011

Status

approved