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, 16, -34, -119, 272, 544, -1360, -1700, 4760, 3808, -12376, -6188, 24752, 7072, -38896, -4862, 48620, 0, -48620, 4862, 38896, -7072, -24752, 6188, 12376, -3808, -4760, 1700, 1360, -544, -272, 119, 34, -16, -2, 1).
FORMULA
a(n) = 2*a(n-1) +16*a(n-2) -34*a(n-3) -119*a(n-4) +272*a(n-5) +544*a(n-6) -1360*a(n-7) -1700*a(n-8) +4760*a(n-9) +3808*a(n-10) -12376*a(n-11) -6188*a(n-12) +24752*a(n-13) +7072*a(n-14) -38896*a(n-15) -4862*a(n-16) +48620*a(n-17) -48620*a(n-19) +4862*a(n-20) +38896*a(n-21) -7072*a(n-22) -24752*a(n-23) +6188*a(n-24) +12376*a(n-25) -3808*a(n-26) -4760*a(n-27) +1700*a(n-28) +1360*a(n-29) -544*a(n-30) -272*a(n-31) +119*a(n-32) +34*a(n-33) -16*a(n-34) -2*a(n-35) +a(n-36). - 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 * (n+12)^2 / 2174327193600 for n even.
a(n) = (n+3)^2 * (n+5)^4 * (n+7)^4 * (n+9)^4 * (n+11)^3 * (n+13) / 2174327193600 for n odd. (End)
From Amiram Eldar, Jun 28 2026: (Start)
Sum_{n>=1} 1/a(n) = 31400*Pi^4/9 + 9299290*Pi^2/27 + 1400*zeta(3) - 121202025161/32400.
Sum_{n>=1} (-1)^(n+1)/a(n) = 29286070961/32400 - 7400*Pi^4/9 - 2212210*Pi^2/27 - 12600*zeta(3). (End)
EXAMPLE
Some solutions for n=2:
..1..1..1..1..1..1..1..1..1....1..1..1..1..1..1..1..1..0
..1..1..1..1..1..1..1..0..0....1..1..1..1..1..0..1..0..1
..1..1..1..1..1..0..1..0..1....0..1..0..0..0..0..0..0..0
..1..1..1..1..0..0..0..0..0....0..1..0..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+12)^2, (n+3)^2 * (n+5)^4 * (n+7)^4 * (n+9)^4 * (n+11)^3 * (n+13)] / 2174327193600; Array[a, 22] (* Amiram Eldar, Jun 28 2026 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
R. H. Hardin, Dec 11 2011
STATUS
approved