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, 10, -22, -44, 110, 110, -330, -165, 660, 132, -924, 0, 924, -132, -660, 165, 330, -110, -110, 44, 22, -10, -2, 1).
FORMULA
a(n) = 2*a(n-1) +10*a(n-2) -22*a(n-3) -44*a(n-4) +110*a(n-5) +110*a(n-6) -330*a(n-7) -165*a(n-8) +660*a(n-9) +132*a(n-10) -924*a(n-11) +924*a(n-13) -132*a(n-14) -660*a(n-15) +165*a(n-16) +330*a(n-17) -110*a(n-18) -110*a(n-19) +44*a(n-20) +22*a(n-21) -10*a(n-22) -2*a(n-23) +a(n-24). - 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 / 5308416 for n even.
a(n) = (n+3)^2 * (n+5)^4 * (n+7)^4 * (n+9)^2 / 5308416 for n odd. (End)
From Amiram Eldar, Jun 28 2026: (Start)
Sum_{n>=1} 1/a(n) = 117*Pi^4/5 + 4515*Pi^2/2 - 6287361/256.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1683553/256 - 9*Pi^4 - 1155*Pi^2/2. (End)
EXAMPLE
Some solutions for n=3:
..1..1..1..1..0..1....1..1..1..1..1..1....1..1..1..1..0..0....1..0..1..0..0..0
..1..1..0..1..0..1....1..1..0..1..0..0....1..0..1..0..0..0....1..1..0..1..0..1
..1..1..1..1..0..1....0..1..0..1..0..1....1..1..0..0..0..0....1..0..1..0..0..0
..1..1..0..1..0..1....0..0..0..0..0..0....1..0..0..0..0..0....1..0..0..0..0..0
..1..1..0..0..0..0....0..1..0..1..0..1....0..1..0..0..0..0....0..0..0..0..0..0
MATHEMATICA
a[n_] := If[EvenQ[n], (n+4)^4 * (n+6)^4 * (n+8)^4, (n+3)^2 * (n+5)^4 * (n+7)^4 * (n+9)^2] / 5308416; Array[a, 26] (* Amiram Eldar, Jun 28 2026 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
R. H. Hardin, Dec 11 2011
STATUS
approved