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A083686 a(0) = 8; for n>0, a(n) = 2*a(n-1) - 1.

%I A083686 #48 Apr 02 2024 11:07:18
%S A083686 8,15,29,57,113,225,449,897,1793,3585,7169,14337,28673,57345,114689,
%T A083686 229377,458753,917505,1835009,3670017,7340033,14680065,29360129,
%U A083686 58720257,117440513,234881025,469762049,939524097,1879048193,3758096385,7516192769,15032385537
%N A083686 a(0) = 8; for n>0, a(n) = 2*a(n-1) - 1.
%C A083686 An Engel expansion of 2/7 to the base 2 as defined in A181565, with the associated series expansion 2/7 = 2/8 + 2^2/(8*15) + 2^3/(8*15*29) + 2^4/(8*15*29*57) + ... . - _Peter Bala_, Oct 29 2013
%C A083686 The initial 8 is the only cube in this sequence. - _Antti Karttunen_, Sep 24 2023
%H A083686 Vincenzo Librandi, <a href="/A083686/b083686.txt">Table of n, a(n) for n = 0..1000</a>
%H A083686 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A083686 a(n) = 7*2^n + 1. - David Brotherton (dbroth01(AT)aol.com), Jul 29 2003
%F A083686 a(n) = 3*a(n-1) - 2*a(n-2), n>1. - _Vincenzo Librandi_, Nov 03 2011
%F A083686 G.f.: (8-9*x) / ((1-x)*(1-2*x)). - _Colin Barker_, Sep 20 2016
%F A083686 E.g.f.: exp(x)*(1 + 7*exp(x)). - _Stefano Spezia_, Oct 08 2022
%F A083686 For n >= 0, A005940(a(n)) = A030078(1+n). - _Antti Karttunen_, Sep 24 2023
%t A083686 7*2^Range[0, 50] + 1 (* _Paolo Xausa_, Apr 02 2024 *)
%o A083686 (Magma) [7*2^n+1 : n in [0..30]]; // _Vincenzo Librandi_, Nov 03 2011
%o A083686 (PARI) Vec((8-9*x)/((1-x)*(1-2*x)) + O(x^40)) \\ _Colin Barker_, Sep 20 2016
%o A083686 (PARI) a(n)=7<<n + 1 \\ _Charles R Greathouse IV_, Sep 20 2016
%Y A083686 Cf. A020737, A083575, A083683, A083705, A168596, A181565, A195744.
%Y A083686 Cf. A005940, A030078.
%K A083686 nonn,easy
%O A083686 0,1
%A A083686 _N. J. A. Sloane_, Jun 15 2003