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 A007055 Let S denote the palindromes in the language {0,1}*; a(n) = number of words of length n in the language SS. (Formerly M1124) 9
 1, 2, 4, 8, 16, 32, 52, 100, 160, 260, 424, 684, 1036, 1640, 2552, 3728, 5920, 8672, 13408, 19420, 30136, 42736, 66840, 94164, 145900, 204632, 317776, 441764, 685232, 950216, 1469632, 2031556, 3139360, 4323888, 6675904, 9174400, 14139496, 19398584, 29864888, 40891040, 62882680, 85983152 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of words in {0,1}* of length n that are rotations of their reversals. - David W. Wilson, Jan 01 2012 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Chuan Guo, J. Shallit, A. M. Shur, On the Combinatorics of Palindromes and Antipalindromes, arXiv preprint arXiv:1503.09112 [cs.FL], 2015. R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. FORMULA a(n) = A187272(n) - Sum_{d|n, d expand(simplify( (n/4)*a^(n/2)*( (1+sqrt(a))^2+ (-1)^n*(1-sqrt(a))^2 ) )); # A007055, A007056, A007057, A007058 F:=(b, n)-> if n=0 then 1 else expand(simplify( add( f(d)*R(b, n/d), d in divisors(n) ) )); fi; # A007055: [seq(F(2, n), n=0..60)]; MATHEMATICA A187272[n_] := A187272[n] = (n/4)*2^(n/2)*((1 + Sqrt)^2 + (-1)^n*(1 - Sqrt)^2) // Round; a[n_ /; n <= 5] := 2^n; a[n_] := a[n] = A187272[n] - Sum[n, EulerPhi[n/d] * a[d], {d, Most[Divisors[n]]}]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *) CROSSREFS Column 2 of A284873. For the nonempty words in the language S, see A057148 and A006995. Cf. A007056-A007058, A023900, A187272-A187275. Sequence in context: A226930 A297702 A306314 * A175951 A072207 A176718 Adjacent sequences:  A007052 A007053 A007054 * A007056 A007057 A007058 KEYWORD nonn AUTHOR N. J. A. Sloane, Mira Bernstein, R. Kemp EXTENSIONS Entry revised by N. J. A. Sloane, Mar 07 2011 STATUS approved

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Last modified May 26 19:44 EDT 2019. Contains 323597 sequences. (Running on oeis4.)