A052548 a(n) = 2^n + 2.

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

Offset: 0

Author

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

Keywords

• easy
• nonn

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 of area n, for n>0, see Beaton, p. 5. - Jonathan Vos Post, Nov 30 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..240
• Nicholas R. Beaton, Philippe Flajolet, and 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).

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.

Programs

Haskell

a052548 = (+ 2) . a000079
a052548_list = iterate ((subtract 2) . (* 2)) 3
-- Reinhard Zumkeller, Sep 05 2015

Magma

[2^n + 2: n in [0..35]]; // Vincenzo Librandi, Apr 29 2011

Maple

spec := [S,{S=Union(Sequence(Union(Z,Z)),Sequence(Z),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

2^Range[0,40]+2 (* Harvey P. Dale, Jun 26 2012 *)

PARI

a(n)=1<Charles R Greathouse IV, Nov 20 2011

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

Extensions

More terms from James Sellers, Jun 06 2000