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 A094536 Number of binary words of length n that are not "bifix-free". 10
 0, 0, 2, 4, 10, 20, 44, 88, 182, 364, 740, 1480, 2980, 5960, 11960, 23920, 47914, 95828, 191804, 383608, 767500, 1535000, 3070568, 6141136, 12283388, 24566776, 49135784, 98271568, 196547560, 393095120, 786199088, 1572398176, 3144813974 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also the number of binary strings of length n that begin with an even length palindrome. (E.g., f(4) = 10 with strings 0000 0001 0010 0011 0110 1001 1100 1101 1110 1111.) - Peter Kagey, Jan 11 2015 The probability that a random, infinite binary string begins with an even-length palindrome is: lim n -> infinity a(n)/2^n ~ 0.7322131597821108... . - Peter Kagey, Jan 26 2015 LINKS Peter Kagey, Table of n, a(n) for n = 0..1000 FORMULA a(n) = 2^n - A003000(n). Let b(0)=1; b(n) = 2*b(n-1) - 1/2*(1+(-1)^n)*b([n/2]); a(n) = 2^n - b(n). - Farideh Firoozbakht, Jun 10 2004 a(0) = 0; a(1) = 0; a(2*n+1) = 2*a(2*n); a(2*n) = 2*a(2*n-1) + 2^n - a(n). - Peter Kagey, Jan 11 2015 G.f. g(x) satisfies (1-2*x)*g(x) = 2*x^2/(1-2*x^2) - g(x^2). - Robert Israel, Jan 12 2015 MAPLE a[0]:= 0: for n from 1 to 100 do if n::odd then a[n]:= 2*a[n-1] else a[n]:= 2*a[n-1] + 2^(n/2) - a[n/2] fi od: seq(a[i], i=0..100); # Robert Israel, Jan 12 2015 MATHEMATICA b[0]=1; b[n_]:=b[n]=2*b[n-1]-(1+(-1)^n)/2*b[Floor[n/2]]; a[n_]:=2^n-b[n]; Table[a[n], {n, 0, 34}] PROG (Ruby) s = [0, 0] (2..N).each { |n| s << 2 * s[-1] + (n.odd? ? 0 : 2**(n/2) - s[n/2]) } CROSSREFS See A003000 for much more information. Cf. A094537. A254128(n) gives the number of binary strings of length n that begin with an odd-length palindrome. Sequence in context: A325508 A238439 A121880 * A323445 A307768 A297183 Adjacent sequences:  A094533 A094534 A094535 * A094537 A094538 A094539 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Jun 06 2004 EXTENSIONS More terms from Farideh Firoozbakht, Jun 10 2004 Corrected by Don Rogers (donrogers42(AT)aol.com), Feb 15 2005 STATUS approved

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Last modified May 22 17:42 EDT 2022. Contains 353957 sequences. (Running on oeis4.)