A057198 a(n) = (5*3^(n-1)+1)/2.

3, 8, 23, 68, 203, 608, 1823, 5468, 16403, 49208, 147623, 442868, 1328603, 3985808, 11957423, 35872268, 107616803, 322850408, 968551223, 2905653668, 8716961003, 26150883008, 78452649023, 235357947068, 706073841203, 2118221523608

Offset: 1

Author

Colin Mallows and N. J. A. Sloane, Sep 16 2000

Keywords

• nonn
• easy

Comments

It appears that if s(n) is a first-order rational sequence of the form s(0)=4, s(n) = (2*s(n-1)+1)/(s(n-1)+2), n > 0, then s(n) = a(n)/(a(n)-1), n > 0.

Examples

			G.f. = 3*x + 8*x^2 + 23*x^3 + 68*x^4 + 203*x^5 + 608*x^6 + 1823*x^7 + 5468*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Related to A046901.

Equals A060816 + 1.

Cf. A135423 (bisection), A191450 (2nd row).

Programs

Magma

[(5*3^(n-1) + 1)/2: n in [1..30]]; // Vincenzo Librandi, Oct 11 2012

Mathematica

Table[(5*3^(n-1) + 1)/2, {n, 30}] (* T. D. Noe, Oct 11 2012 *)

PARI

a(n)=(5*3^(n-1)+1)/2 \\ Charles R Greathouse IV, Oct 11 2012

Formula

a(n+1) = 3*a(n) - 1 for n > 1. - Reinhard Zumkeller, Jan 22 2011

G.f.: (5/2)*U(0) where U(k) = 1 + 2/(5*3^k + 5*3^k/(1 - 30*x*3^k/(15*x*3^k - 1/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012

E.g.f.: (5/2)*U(0) where U(k) = 1 + 2/(5*3^k + 5*3^k/(1 - 30*x*3^k/(15*x*3^k - (k+1)/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012

G.f.: x*(3-4*x) / ( (3*x-1)*(x-1) ). - R. J. Mathar, Jan 25 2015

E.g.f.: (5*exp(3*x) + 3*exp(x) - 8)/6. - Stefano Spezia, Aug 28 2023

Extensions

Incorrect zeroth term removed by Jon Perry, Oct 11 2012