A052919 a(n) = 1 + 2*3^(n-1) with a(0)=2.

2, 3, 7, 19, 55, 163, 487, 1459, 4375, 13123, 39367, 118099, 354295, 1062883, 3188647, 9565939, 28697815, 86093443, 258280327, 774840979, 2324522935, 6973568803, 20920706407, 62762119219, 188286357655, 564859072963

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

It appears that if s(n) is a first order rational sequence of the form s(1)=3, s(n) = (2*s(n-1)+1)/(s(n-1)+2), n > 1, then s(n) = a(n)/(a(n)-2).

The binomial transform is 2, 5, 15, 51, 187, ...A007581 without the leading term. - R. J. Mathar, Apr 07 2022

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.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 902
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Programs

GAP

Concatenation([2], List([1..30], n-> 1 + 2*3^(n-1) )); # G. C. Greubel, Oct 16 2019

Magma

I:=[2, 3, 7]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

Maple

spec := [S,{S=Union(Sequence(Prod(Sequence(Z),Union(Z,Z))),Sequence(Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(`if`(n=0, 2, 1 + 2*3^(n-1)), n=0..30); # G. C. Greubel, Oct 16 2019

Mathematica

Join[{2},Table[2*(3^n+1)-1,{n,0,30}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
CoefficientList[Series[(2-5*x+x^2)/((1-x)*(1-3*x)),{x,0,40}],x] (* Vincenzo Librandi, Jun 22 2012 *)
LinearRecurrence[{4,-3},{2,3,7},30] (* Harvey P. Dale, Dec 12 2017 *)

PARI

vector(31, n, if(n==1, 2, 1+ 2*3^(n-2))) \\ G. C. Greubel, Oct 16 2019

Sage

[2]+[1+2*3^(n-1) for n in (1..30)] # G. C. Greubel, Oct 16 2019

Formula

a(n) = 1 + 2*3^(n-1) for n > 0 with a(0) = 2.

G.f.: (2 - 5*x + x^2)/((1-x)*(1-3*x)).

a(n) = 4*a(n-1) - 3*a(n-2), with a(0)=2, a(1)=3, a(2)=7.

a(0) = 2 and a(n) = A100702(n) for n >= 1. - Omar E. Pol, Mar 02 2012

a(n) = A104743(n) - A104743(n-1). - J. M. Bergot, Jun 07 2013

Extensions

More terms from James Sellers, Jun 05 2000