A202098 Number of (n+2) X 8 binary arrays avoiding patterns 001 and 011 in rows and columns.

5625, 50625, 275625, 1500625, 6002500, 24010000, 77792400, 252047376, 700131600, 1944810000, 4802490000, 11859210000, 26683222500, 60037250625, 125262905625, 261351000625, 512247961225, 1004006004001, 1866953313225, 3471607400625, 6171746490000, 10971993760000, 18762758560000

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 28 2026

Links

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

Index entries for linear recurrences with constant coefficients, signature (2, 14, -30, -90, 210, 350, -910, -910, 2730, 1638, -6006, -2002, 10010, 1430, -12870, 0, 12870, -1430, -10010, 2002, 6006, -1638, -2730, 910, 910, -350, -210, 90, 30, -14, -2, 1).

Formula

a(n) = 2*a(n-1) +14*a(n-2) -30*a(n-3) -90*a(n-4) +210*a(n-5) +350*a(n-6) -910*a(n-7) -910*a(n-8) +2730*a(n-9) +1638*a(n-10) -6006*a(n-11) -2002*a(n-12) +10010*a(n-13) +1430*a(n-14) -12870*a(n-15) +12870*a(n-17) -1430*a(n-18) -10010*a(n-19) +2002*a(n-20) +6006*a(n-21) -1638*a(n-22) -2730*a(n-23) +910*a(n-24) +910*a(n-25) -350*a(n-26) -210*a(n-27) +90*a(n-28) +30*a(n-29) -14*a(n-30) -2*a(n-31) +a(n-32). - Proved by Christian Krause, Jun 28 2026

From Christian Krause, Jun 28 2026: (Start)

a(n) = (n+4)^4 * (n+6)^4 * (n+8)^4 * (n+10)^4 / 21743271936 for n even.

a(n) = (n+3)^2 * (n+5)^4 * (n+7)^4 * (n+9)^4 * (n+11)^2 / 21743271936 for n odd. (End)

From Amiram Eldar, Jun 28 2026: (Start)

Sum_{n>=1} 1/a(n) = 33664*Pi^4/45 + 2002000*Pi^2/27 - 40737104606/50625.

Sum_{n>=1} (-1)^(n+1)/a(n) = 11285196806/50625 - 1664*Pi^4/9 - 560560*Pi^2/27. (End)

Example

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

Mathematica

a[n_] := If[EvenQ[n], (n+4)^4 * (n+6)^4 * (n+8)^4 * (n+10)^4, (n+3)^2 * (n+5)^4 * (n+7)^4 * (n+9)^4 * (n+11)^2] / 21743271936; Array[a, 23] (* Amiram Eldar, Jun 28 2026 *)

Crossrefs

Column 6 of A202100.

Cf. A202093.

Sequence in context: A237507 A198164 A058003 * A247261 A203831 A014799

Adjacent sequences: A202095 A202096 A202097 * A202099 A202100 A202101

Keyword

nonn,easy,changed

Author

R. H. Hardin, Dec 11 2011

Status

approved