A168607 a(n) = 3^n + 2.

3, 5, 11, 29, 83, 245, 731, 2189, 6563, 19685, 59051, 177149, 531443, 1594325, 4782971, 14348909, 43046723, 129140165, 387420491, 1162261469, 3486784403, 10460353205, 31381059611, 94143178829, 282429536483, 847288609445

Offset: 0

Author

Vincenzo Librandi, Dec 01 2009

Keywords

• nonn
• easy

Comments

Second bisection is A134752.

It appears that if s(n) is a first order rational sequence of the form s(1)=5, s(n)= (2*s(n-1)+1)/(s(n-1)+2),n>1, then s(n)= a(n)/(a(n)-4), n>1. - Gary Detlefs, Nov 16 2010

Mahler exhibits this sequence with n>=1 as a proof that there exists an infinite number of x coprime to 3, such that x belongs to A125293 and x^2 belongs to A005836. - Michel Marcus, Nov 12 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.
• Kurt Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 99-106.
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A008776 (2*3^n), A005051 (8*3^n), A034472 (3^n+1), A000244 (powers of 3), A024023 (3^n-1), A168609 (3^n+4), A168610 (3^n+5), A134752 (3^(2*n-1)+2).

Programs

Magma

[3^n+2: n in [0..30]];

Maple

A168607:=n->3^n + 2; seq(A168607(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

Mathematica

CoefficientList[Series[(3 - 7 x)/((1-x) (1-3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
NestList[3 # - 4 & , 3, 25] (* Bruno Berselli, Feb 06 2013 *)

PARI

a(n)=3^n+2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 3*a(n-1) - 4, a(0) = 3.

a(n+1) - a(n) = A008776(n).

a(n+2) - a(n) = A005051(n).

a(n) = A034472(n)+1 = A000244(n)+2 = A024023(n)+3 = A168609(n)-2 = A168610(n)-3.

G.f.: (3 - 7*x)/((1 - x)*(1 - 3*x)).

a(n) = 4*a(n-1) - 3*a(n-2), a(0) = 3, a(1) = 5. - Vincenzo Librandi, Feb 06 2013

E.g.f.: exp(3*x) + 2*exp(x). - Elmo R. Oliveira, Nov 09 2023

Extensions

Edited by Klaus Brockhaus, Apr 13 2010

Further edited by N. J. A. Sloane, Aug 10 2010