A083713 a(n) = (8^n - 1)*3/7.

0, 3, 27, 219, 1755, 14043, 112347, 898779, 7190235, 57521883, 460175067, 3681400539, 29451204315, 235609634523, 1884877076187, 15079016609499, 120632132875995, 965057063007963, 7720456504063707, 61763652032509659

Offset: 0

Author

Klaus Brockhaus, Jun 14 2003

Keywords

• nonn
• easy

Comments

Fixed points of the mapping defined by A067585. In binary these numbers show a regular pattern: 0, 11, 11011, 11011011, 11011011011, etc.

From Reinhard Zumkeller, Feb 22 2010: (Start)

a(n) = A173593(6*n-5) for n > 0:

terms of A173593 beginning and ending with digits '11' in binary representation;

for n > 0: a(n) = A033129(3*n-1); a(n) - a(n-1) = A103333(n). (End)

Examples

			From _Zerinvary Lajos_, Jan 14 2007: (Start)
Octal..........decimal:
0....................0
3....................3
33..................27
333................219
3333..............1755
33333............14043
333333..........112347
3333333.........898779
33333333.......7190235
333333333.....57521883
3333333333...460175067
etc. (End)
		

Links

• Index entries for linear recurrences with constant coefficients, signature (9,-8).

Crossrefs

Cf. A067585, A023001.

Programs

Mathematica

(3/7)(8^Range[0,20]-1) (* or *) LinearRecurrence[{9,-8},{0,3},30] (* or *) NestList[8#+3&,0,30] (* Harvey P. Dale, Jun 06 2013 *)

PARI

a(n)=(8^n-1)*3/7 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 3*A023001(n).

Recursion: a(0) = 0, a(n+1) = (((a(n)*2)*2+1)*2+1).

a(n) = 8*a(n-1) + 3 (with a(0)=0). - Vincenzo Librandi, Aug 08 2010

a(0)=0, a(1)=3, a(n) = 9*a(n-1) - 8*a(n-2). - Harvey P. Dale, Jun 06 2013

From Stefano Spezia, Feb 23 2025: (Start)

G.f.: 3*x/((1 - x)*(1 - 8*x)).

E.g.f.: 3*exp(x)*(exp(7*x) - 1)/7. (End)