A140660 a(n) = 3*4^n + 1.

4, 13, 49, 193, 769, 3073, 12289, 49153, 196609, 786433, 3145729, 12582913, 50331649, 201326593, 805306369, 3221225473, 12884901889, 51539607553, 206158430209, 824633720833, 3298534883329, 13194139533313, 52776558133249

Offset: 0

Author

Paul Curtz, Jul 10 2008

Keywords

• nonn

Comments

An Engel expansion of 4/3 to the base 4 as defined in A181565, with the associated series expansion 4/3 = 4/4 + 4^2/(4*13) + 4^3/(4*13*49) + 4^4/(4*13*49*193) + .... Cf. A199115. - Peter Bala, Oct 29 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (5,-4).

Crossrefs

Cf. A181565, A199115.

Programs

Magma

[3*4^n+1: n in [0..30] ]; // Vincenzo Librandi, May 23 2011

Mathematica

LinearRecurrence[{5,-4}, {4,13}, 50] (* or *) CoefficientList[Series[ (7*x-4)/((1-x)*(4*x-1)), {x,0,50}], x] (* G. C. Greubel, Sep 15 2017 *)

PARI

x='x+O('x^50); Vec((7*x-4)/((1-x)*(4*x-1))) \\ G. C. Greubel, Sep 15 2017

Formula

a(n) = A002001(n+1) + 1.

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

First differences: a(n+1) - a(n) = A002063(n).

a(n+k) - a(n) = 3*(4^k - 1)*A000302(n) = 9*A002450(k)*A000302(n).

a(n) = A140529(n) - A096045(n).

O.g.f.: (7*x - 4)/((1 - x)*(4*x - 1)). - R. J. Mathar, Jul 14 2008

From G. C. Greubel, Sep 15 2017: (Start)

E.g.f.: 3*exp(4*x) + exp(x).

a(n) = 5*a(n-1) - 4*a(n-2). (End)

Extensions

Edited and extended R. J. Mathar, Jul 14 2008