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 A079583 a(n) = 3*2^n - n - 2. 17
 1, 3, 8, 19, 42, 89, 184, 375, 758, 1525, 3060, 6131, 12274, 24561, 49136, 98287, 196590, 393197, 786412, 1572843, 3145706, 6291433, 12582888, 25165799, 50331622, 100663269, 201326564, 402653155, 805306338, 1610612705, 3221225440 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row sums of A132110. - Gary W. Adamson, Aug 09 2007 Consider the infinite sequence of strings x(1) = a, x(2) = aba, x(3) = ababbaba, ..., where x(n+1) = x(n).b^{n+1}.x(n), for n >= 1. Each x(n), for n >= 2, has borders x(1), x(2), ..., x(n-1), none of which cover x(n). The length of x(n+1) is 3*2^n-n-2. - William F. Smyth, Feb 29 2012 Number of edges in the rooted tree g[n] (n>=0) defined recursively in the following manner: denoting by P[n] the path on n vertices, we define g[0] =P[2] while g[n] (n>=1) is the tree obtained by identifying the roots of 2 copies of g[n-1] and one of the end-vertices of P[n+1]; the root of g[n] is defined to be the other end-vertex of P[n+1]. Roughly speaking, g[4], for example, is obtained from the planted full binary tree of height 5 by replacing the edges at the levels 1,2,3,4 with paths of lengths 4, 3, 2, and 1, respectively. - Emeric Deutsch, Aug 08 2013 REFERENCES T. Flouri, C. S. Iliopoulos, T. Kociumaka, S. P. Pissis, S. J. Puglisi, W. F. Smyth, W. Tyczynski, New and efficient approaches to the quasiperiodic characterization of a string, Proc. Prague Stringology Conf., 2012, 75-88. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Tomas Flouri, Costas S. Iliopoulos, Solon P. Pissis, W. F. Smyth, On approximate string covering (draft, 2012). Index entries for linear recurrences with constant coefficients, signature (4,-5,2). FORMULA a(0)=1, a(n) = 2*a(n-1) + n; Binomial transform of [1, 2, 3, 3, 3, ...]. - Gary W. Adamson, Aug 09 2007 G.f.: (x^2-x+1)/((1-2*x)*(1-x)^2) = 3*U(0)x, where U(k) = 1 - (k+2)/(3*2^k - 18*x*4^k/(6*x*2^k - (k+2)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Jul 04 2012 a(n) = (A227712(n) - 1)/3 - Emeric Deutsch, Feb 18 2016 a(n) = A007283(n) - n - 2. - Miquel Cerda, Aug 07 2016 a(n) = A000225(n) + A000325(n). - Miquel Cerda, Aug 08 2016 MATHEMATICA lst={}; Do[AppendTo[lst, 3*2^n-n-2], {n, 0, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *) LinearRecurrence[{4, -5, 2}, {1, 3, 8}, 40] (* Vincenzo Librandi, Jun 23 2012 *) PROG (PARI) a(n)=3<

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