A213123 Number of binary arrays of length 2*n+5 with fewer than n ones in any length 2n subsequence (=less than 50% duty cycle).

1, 26, 212, 1198, 5842, 26630, 116992, 502492, 2126238, 8903350, 36998056, 152862180, 628749892, 2576996188, 10531805664, 42940549576, 174734720374, 709858318486, 2879728611544, 11668224303796, 47228199967804

Offset: 1

Author

R. H. Hardin, Jun 05 2012

Keywords

• nonn

Examples

			Some solutions for n=3:
  0  1  1  0  0  1  1  0  0  0  1  0  1  0  1  0
  1  0  0  0  1  0  0  0  1  1  0  0  0  0  0  0
  0  1  0  1  1  0  0  0  0  1  1  0  0  0  0  0
  0  0  0  0  0  0  0  0  1  0  0  1  0  0  0  0
  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0
  0  0  1  0  0  0  0  0  0  0  0  0  0  1  0  0
  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  0
  0  1  0  1  0  0  0  1  1  0  0  0  0  0  0  0
  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0  1
  0  0  0  0  0  0  0  0  1  1  0  0  1  0  0  0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210

Crossrefs

Row 6 of A213118.

Programs

Maple

#verified first terms (holds for all n<=210).
with(gfun): A213123:= rectoproc({a(3)=212, a(4)=1198, n*(33*n^2-213*n+340)*a(n) = 2*(132*n^3-951*n^2+2029*n-1120)*a(n-1) - 8*(2*n-7)*(33*n^2-147*n+160)*a(n-2)},a(n),remember): 1,26,seq(A213123(n),n=3..20); A213123(210); # Vaclav Kotesovec, Nov 20 2012

Formula

Empirical (for n>=5): n*(33*n^2 - 213*n + 340)*a(n) = 2*(132*n^3 - 951*n^2 + 2029*n - 1120)*a(n-1) - 8*(2*n-7)*(33*n^2 - 147*n + 160)*a(n-2). - Vaclav Kotesovec, Nov 20 2012

Empirical (for n>=3): a(n) = 4^(n+2) - 42*(33*n^2 - 71*n + 32) * C(2*n - 5, n - 3) / ((n-1)*n). - Vaclav Kotesovec, Nov 20 2012