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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
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OFFSET
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0,1
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COMMENTS
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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 Plewe, Dec 11 2007
a(n) = A058896(n)/A000918(n). [From Reinhard Zumkeller, Feb 14 2009]
For n>0: a(n) = A173786(n,1) and a(n)*A000918(n)=A028399(2*n). [From Reinhard Zumkeller, Feb 28 2010]
a(n) = 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. [From Jonathan Vos Post, Nov 30 2010]
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REFERENCES
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"Sieves", Popular Computing (Calabasas, CA), Vol. 2 (No. 13, Apr 1974), pp. 6-7; sieve #6 (K=2).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..240
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 485
Index entries for sequences generated by sieves
Eric Weisstein's World of Mathematics, Bertrand's Postulate
Nicholas R. Beaton, Philippe Flajolet, Anthony J. Guttmann, The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics, Nov 29, 2010.
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FORMULA
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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(0)=3, a(n)=2*a(n-1)-2 [From Vincenzo Librandi, Aug 06 2010]
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EXAMPLE
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a(1)=2*3-2=4; a(2)=2*4-2=6; a(3)=2*6-2=10; a(4)=2*10-2=18 [From Vincenzo Librandi, Aug 06 2010]
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MAPLE
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spec := [S, {S=Union(Sequence(Union(Z, Z)), Sequence(Z), Sequence(Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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a=3; lst={a}; Do[a=2*a-2; AppendTo[lst, a], {n, 0, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Dec 25 2008]
2^Range[0, 40]+2 (* Harvey P. Dale, Jun 26 2012 *)
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PROG
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(MAGMA) [2^n + 2: n in [0..35]]; // Vincenzo Librandi, Apr 29 2011
(PARI) a(n)=1<<n+2 \\ Charles R Greathouse IV, Nov 20 2011
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CROSSREFS
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Apart from initial term, same as A056469.
Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774, A079004, A058481 [From Vladimir Joseph Stephan Orlovsky, Dec 25 2008]
Sequence in context: A068922 A032408 A018908 * A103049 A103016 A061032
Adjacent sequences: A052545 A052546 A052547 * A052549 A052550 A052551
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KEYWORD
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easy,nonn,changed
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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More terms from James A. Sellers, Jun 06 2000
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STATUS
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approved
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