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URL: https://oeis.org/A052919

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A052919
a(n) = 1 + 2*3^(n-1) with a(0)=2.
13
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, 1694577218887, 5083731656659, 15251194969975
OFFSET
0,1
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
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
From Elmo R. Oliveira, Sep 14 2025: (Start)
E.g.f.: (1 + 3*exp(x) + 2*exp(3*x))/3.
a(n) = 1 + sqrt(A270473(n)).
a(n) = A384853(n+2)/3 for n >= 1. (End)
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 *)
PROG
(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
(PARI) vector(31, n, if(n==1, 2, 1+ 2*3^(n-2))) \\ G. C. Greubel, Oct 16 2019
(SageMath) [2]+[1+2*3^(n-1) for n in (1..30)] # G. C. Greubel, Oct 16 2019
(GAP) Concatenation([2], List([1..30], n-> 1 + 2*3^(n-1) )); # G. C. Greubel, Oct 16 2019
CROSSREFS
KEYWORD
easy,nonn
EXTENSIONS
More terms from James Sellers, Jun 05 2000
STATUS
approved