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, 12, -26, -65, 156, 208, -572, -429, 1430, 572, -2574, -429, 3432, 0, -3432, 429, 2574, -572, -1430, 429, 572, -208, -156, 65, 26, -12, -2, 1).
FORMULA
a(n) = 2*a(n-1) +12*a(n-2) -26*a(n-3) -65*a(n-4) +156*a(n-5) +208*a(n-6) -572*a(n-7) -429*a(n-8) +1430*a(n-9) +572*a(n-10) -2574*a(n-11) -429*a(n-12) +3432*a(n-13) -3432*a(n-15) +429*a(n-16) +2574*a(n-17) -572*a(n-18) -1430*a(n-19) +429*a(n-20) +572*a(n-21) -208*a(n-22) -156*a(n-23) +65*a(n-24) +26*a(n-25) -12*a(n-26) -2*a(n-27) +a(n-28). - 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)^2 / 339738624 for n even.
a(n) = (n+3)^2 * (n+5)^4 * (n+7)^4 * (n+9)^3 * (n+11) / 339738624 for n odd. (End)
From Amiram Eldar, Jun 28 2026: (Start)
Sum_{n>=1} 1/a(n) = 112*Pi^4 + 33110*Pi^2/3 - 240*zeta(3) - 430375889/3600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 123627329/3600 + 1680*zeta(3) - 176*Pi^4/5 - 10010*Pi^2/3. (End)
EXAMPLE
Some solutions for n=2:
..1..1..1..1..0..0..0....1..1..1..1..0..0..0....1..1..1..1..1..1..1
..1..1..0..1..0..1..0....1..1..1..1..1..0..0....1..1..0..1..0..1..0
..0..1..0..1..0..0..0....1..1..1..1..0..0..0....1..0..1..0..1..0..1
..1..0..0..0..0..0..0....1..1..0..0..0..0..0....1..1..0..1..0..0..0
MATHEMATICA
a[n_] := If[EvenQ[n], (n+4)^4 * (n+6)^4 * (n+8)^4 * (n+10)^2, (n+3)^2 * (n+5)^4 * (n+7)^4 * (n+9)^3 * (n+11)] / 339738624; Array[a, 25] (* Amiram Eldar, Jun 28 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin, Dec 11 2011
STATUS
approved