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 A052548 a(n) = 2^n + 2. 48
 3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The most "compact" sequence that satisfies Bertrand's Postulate. Begin with a(1) = 3 = n, then 2n - 2 = 4 = n_1, 2n_1 - 2 = 6 = n_2, 2n_2 - 2 = 10, etc. = a(n), hence there is guaranteed to be at least one prime between successive members of the sequence. - Andrew S. Plewe, Dec 11 2007 Number of 2-sided prudent polygons with area n, for n>0, proved by Beaton, p. 5. Prudent polygons are prudent walks that return to a point adjacent to their starting point. - Jonathan Vos Post, Nov 30 2010 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..240 Nicholas R. Beaton, Philippe Flajolet, Anthony J. Guttmann, The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics, arXiv:1011.6195 [math.CO], Nov 29, 2010. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 485 Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #6 with K=2. [Annotated and scanned copy] Eric Weisstein's World of Mathematics, Bertrand's Postulate Index entries for linear recurrences with constant coefficients, signature (3,-2) FORMULA G.f.: (3-5*x)/((1-2*x)*(1-x)) = (3-5*x)/(1 - 3*x + 2*x^2) = 2/(1-x) + 1/(1-2*x). a(0)=3, a(1)=4, a(n) = 3*a(n-1) - 2*a(n-2). a(n) = A058896(n)/A000918(n), for n>0. - Reinhard Zumkeller, Feb 14 2009 a(n) = A173786(n,1), for n>0. - Reinhard Zumkeller, Feb 28 2010 a(n)*A000918(n) = A028399(2*n), for n>0. - Reinhard Zumkeller, Feb 28 2010 a(0)=3, a(n) = 2*a(n-1) - 2. - Vincenzo Librandi, Aug 06 2010 E.g.f.: (2 + exp(x))*exp(x). - Ilya Gutkovskiy, Aug 16 2016 MAPLE spec := [S, {S=Union(Sequence(Union(Z, Z)), Sequence(Z), Sequence(Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA a=3; lst={a}; Do[a=2*a-2; AppendTo[lst, a], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *) 2^Range[0, 40]+2 (* Harvey P. Dale, Jun 26 2012 *) PROG (MAGMA) [2^n + 2: n in [0..35]]; // Vincenzo Librandi, Apr 29 2011 (PARI) a(n)=1<

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Last modified October 21 16:25 EDT 2019. Contains 328302 sequences. (Running on oeis4.)