A202201 - OEIS

%I #24 Jun 28 2026 17:41:29

%S 2430,11880,44550,138996,378378,926640,2084940,4375800,8664084,

%T 16325712,29476980,51279480,86337900,141210432,225054126,350430300,

%U 534298050,799227000,1174863690,1699689420,2423110950,3407929200,4733235000,6497785008,8823915144,11862053280,15795897480,20848330800

%N Number of (n+2) X 9 binary arrays avoiding patterns 001 and 101 in rows and columns.

%C Part of the family a(n) = 2*w*(n+2)*C(n+w,w-1) for width-w binary arrays avoiding patterns 001 and 101 (A202195-A202201 for w=3..9). - _Christian Krause_, Jun 25 2026

%H R. H. Hardin, <a href="/A202201/b202201.txt">Table of n, a(n) for n = 1..210</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F a(n) = (n+9)*(n+8)*(n+7)*(n+6)*(n+5)*(n+4)*(n+3)*(n+2)^2/2240. [proved by _Christian Krause_, Jun 25 2026]

%F From _Colin Barker_, May 27 2018: (Start)

%F G.f.: 18*x*(135 - 690*x + 1950*x^2 - 3528*x^3 + 4326*x^4 - 3660*x^5 + 2115*x^6 - 800*x^7 + 179*x^8 - 18*x^9) / (1 - x)^10.

%F a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 10. (End)

%F From _Amiram Eldar_, Jun 28 2026: (Start)

%F Sum_{n>=1} 1/a(n) = 2*Pi^2/27 - 24163/33075.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/27 + 1370464/99225 - 19328*log(2)/945. (End)

%e Some solutions for n=1:

%e ..1..0..0..0..0..0..0..0..0....1..1..1..1..1..1..1..0..0

%e ..1..1..1..1..1..1..1..1..1....1..1..1..1..1..1..1..1..1

%e ..1..1..1..1..0..0..0..0..0....1..1..1..1..0..0..0..0..0

%t a[n_] := (n+9)*(n+8)*(n+7)*(n+6)*(n+5)*(n+4)*(n+3)*(n+2)^2/2240; Array[a, 28] (* _Amiram Eldar_, Jun 28 2026 *)

%Y Column 7 of A202202.

%Y Cf. A202195.

%K nonn,easy

%O 1,1

%A _R. H. Hardin_, Dec 14 2011