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

%I A083713 #29 Feb 23 2025 11:20:01
%S A083713 0,3,27,219,1755,14043,112347,898779,7190235,57521883,460175067,
%T A083713 3681400539,29451204315,235609634523,1884877076187,15079016609499,
%U A083713 120632132875995,965057063007963,7720456504063707,61763652032509659
%N A083713 a(n) = (8^n - 1)*3/7.
%C A083713 Fixed points of the mapping defined by A067585. In binary these numbers show a regular pattern: 0, 11, 11011, 11011011, 11011011011, etc.
%C A083713 From _Reinhard Zumkeller_, Feb 22 2010: (Start)
%C A083713 a(n) = A173593(6*n-5) for n > 0:
%C A083713 terms of A173593 beginning and ending with digits '11' in binary representation;
%C A083713 for n > 0: a(n) = A033129(3*n-1); a(n) - a(n-1) = A103333(n). (End)
%H A083713 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-8).
%F A083713 a(n) = 3*A023001(n).
%F A083713 Recursion: a(0) = 0, a(n+1) = (((a(n)*2)*2+1)*2+1).
%F A083713 a(n) = 8*a(n-1) + 3 (with a(0)=0). - _Vincenzo Librandi_, Aug 08 2010
%F A083713 a(0)=0, a(1)=3, a(n) = 9*a(n-1) - 8*a(n-2). - _Harvey P. Dale_, Jun 06 2013
%F A083713 From _Stefano Spezia_, Feb 23 2025: (Start)
%F A083713 G.f.: 3*x/((1 - x)*(1 - 8*x)).
%F A083713 E.g.f.: 3*exp(x)*(exp(7*x) - 1)/7. (End)
%e A083713 From _Zerinvary Lajos_, Jan 14 2007: (Start)
%e A083713 Octal..........decimal:
%e A083713 0....................0
%e A083713 3....................3
%e A083713 33..................27
%e A083713 333................219
%e A083713 3333..............1755
%e A083713 33333............14043
%e A083713 333333..........112347
%e A083713 3333333.........898779
%e A083713 33333333.......7190235
%e A083713 333333333.....57521883
%e A083713 3333333333...460175067
%e A083713 etc. (End)
%t A083713 (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 *)
%o A083713 (PARI) a(n)=(8^n-1)*3/7 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A083713 Cf. A067585, A023001.
%K A083713 nonn,easy
%O A083713 0,2
%A A083713 _Klaus Brockhaus_, Jun 14 2003