A202199 Number of (n+2) X 7 binary arrays avoiding patterns 001 and 101 in rows and columns.

1176, 4704, 14700, 38808, 90552, 192192, 378378, 700700, 1233232, 2079168, 3378648, 5317872, 8139600, 12155136, 17757894, 25438644, 35802536, 49588000, 67687620, 91171080, 121310280, 159606720, 207821250, 268006284, 342540576, 434166656, 546031024, 681727200, 845341728, 1041503232, 1275434622

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 (8,-28,56,-70,56,-28,8,-1).

Formula

a(n) = 7*(n+7)*(n+6)*(n+5)*(n+4)*(n+3)*(n+2)^2/360. [proved by Christian Krause, Jun 24 2026]

From Colin Barker, May 27 2018: (Start)

G.f.: 14*x*(84 - 336*x + 714*x^2 - 924*x^3 + 756*x^4 - 384*x^5 + 111*x^6 - 14*x^7) / (1 - x)^8.

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n > 8. (End)

From Amiram Eldar, Jun 28 2026: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/140 - 41383/588000.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/280 - 128*log(2)/175 + 277423/588000. (End)

Example

Some solutions for n = 2:
  1  1  1  1  1  1  0    1  1  1  0  0  0  0    0  1  1  1  1  1  1
  1  1  1  1  1  0  0    1  1  1  1  1  1  1    1  1  1  1  1  0  0
  1  0  0  0  0  0  0    1  0  0  0  0  0  0    1  1  1  1  0  0  0
  1  0  0  0  0  0  0    0  0  0  0  0  0  0    1  0  0  0  0  0  0

Mathematica

A202199[n_] := 7*(n+7)*(n+6)*(n+5)*(n+4)*(n+3)*(n+2)^2/360;
Array[A202199, 35] (* Paolo Xausa, Jun 25 2026 *)

Crossrefs

Column 5 of A202202.

Cf. A202195.

Sequence in context: A397461 A204755 A384636 * A233870 A237774 A223340

Adjacent sequences: A202196 A202197 A202198 * A202200 A202201 A202202

Keyword

nonn,easy,changed

Author

R. H. Hardin, Dec 14 2011

Status

approved