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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
Tomas Flouri, Costas S. Iliopoulos, Solon P. Pissis, W. F. Smyth, On approximate string covering (draft, 2012).
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<<n-n-2 \\ Charles R Greathouse IV, Feb 29 2012
(Magma) I:=[1, 3, 8]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 23 2012
CROSSREFS
Cf. A000295, A132110, A227712, A083329 (first differences).
Sequence in context: A002318 A229198 A095681 * A357291 A099050 A065352
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Jan 25 2003
STATUS
approved

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Last modified June 13 09:03 EDT 2024. Contains 373383 sequences. (Running on oeis4.)