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 27 2026
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (2,8,-18,-27,72,48,-168,-42,252,0,-252,42,168,-48,-72,27,18,-8,-2,1).
FORMULA
a(n) = 2*a(n-1) +8*a(n-2) -18*a(n-3) -27*a(n-4) +72*a(n-5) +48*a(n-6) -168*a(n-7) -42*a(n-8) +252*a(n-9) -252*a(n-11) +42*a(n-12) +168*a(n-13) -48*a(n-14) -72*a(n-15) +27*a(n-16) +18*a(n-17) -8*a(n-18) -2*a(n-19) +a(n-20). - Proved by Christian Krause, Jun 27 2026.
From Christian Krause, Jun 27 2026: (Start)
a(n) = (n+4)^4 * (n+6)^4 * (n+8)^2 / 147456 for n even.
a(n) = (n+3)^2 * (n+5)^4 * (n+7)^3 * (n+9) / 147456 for n odd. (End)
From Amiram Eldar, Jun 28 2026: (Start)
Sum_{n>=1} 1/a(n) = 14*Pi^4/5 + 280*Pi^2 + 36*zeta(3) - 443449/144.
Sum_{n>=1} (-1)^(n+1)/a(n) = 246961/144 - 6*Pi^4/5 - 140*Pi^2 - 180*zeta(3). (End)
EXAMPLE
Some solutions for n=4
..1..1..1..0..0....1..1..0..1..0....1..1..1..1..0....1..1..0..0..0
..1..1..1..0..1....1..1..1..0..1....1..1..1..0..0....1..1..0..1..0
..1..1..1..0..0....0..1..0..1..0....1..1..1..1..0....0..1..0..0..0
..0..0..0..0..0....1..1..1..0..1....1..1..0..0..0....0..1..0..0..0
..0..1..0..0..0....0..1..0..1..0....1..1..0..0..0....0..0..0..0..0
..0..0..0..0..0....1..0..1..0..1....1..0..0..0..0....0..1..0..0..0
MATHEMATICA
a[n_] := If[EvenQ[n], (n+4)^4 * (n+6)^4 * (n+8)^2, (n+3)^2 * (n+5)^4 * (n+7)^3 * (n+9)] / 147456; Array[a, 29] (* Amiram Eldar, Jun 28 2026 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
R. H. Hardin, Dec 11 2011
STATUS
approved