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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 sequences generated by sieves

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<<n+2 \\ Charles R Greathouse IV, Nov 20 2011

(Haskell)

a052548 = (+ 2) . a000079

a052548_list = iterate ((subtract 2) . (* 2)) 3

-- Reinhard Zumkeller, Sep 05 2015

CROSSREFS

Apart from initial term, same as A056469.

Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774, A079004, A058481, A100585, A100586, A058896, A000918, A173786.

Cf. also A000079, A000051, A100314.

Sequence in context: A068922 A032408 A018908 * A232268 A103049 A103016

Adjacent sequences:  A052545 A052546 A052547 * A052549 A052550 A052551

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

More terms from James A. Sellers, Jun 06 2000

STATUS

approved

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