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A136412 a(n) = (5*4^n + 1)/3.

%I A136412 #30 Oct 04 2024 12:25:36
%S A136412 2,7,27,107,427,1707,6827,27307,109227,436907,1747627,6990507,
%T A136412 27962027,111848107,447392427,1789569707,7158278827,28633115307,
%U A136412 114532461227,458129844907,1832519379627,7330077518507,29320310074027
%N A136412 a(n) = (5*4^n + 1)/3.
%C A136412 An Engel expansion of 4/5 to the base b := 4/3 as defined in A181565, with the associated series expansion 4/5 = b/2 + b^2/(2*7) + b^3/(2*7*27) + b^4/(2*7*27*107) + .... Cf. A199115 and A140660. - _Peter Bala_, Oct 29 2013
%H A136412 Vincenzo Librandi, <a href="/A136412/b136412.txt">Table of n, a(n) for n = 0..1000</a>
%H A136412 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F A136412 a(n) = 4*a(n-1) - 1.
%F A136412 a(n) = A199115(n)/3.
%F A136412 O.g.f.: (2-3*x)/((1-x)*(1-4*x)). - _R. J. Mathar_, Apr 04 2008
%F A136412 a(n) = 5*a(n-1) - 4*a(n-2). - _Vincenzo Librandi_, Nov 04 2011
%F A136412 E.g.f.: (1/3)*(5*exp(4*x) + exp(x)). - _G. C. Greubel_, Jan 19 2023
%t A136412 LinearRecurrence[{5,-4}, {2,7}, 31] (* _G. C. Greubel_, Jan 19 2023 *)
%o A136412 (Magma) [(5*4^n+1)/3: n in [0..30]]; // _Vincenzo Librandi_, Nov 04 2011
%o A136412 (Haskell)
%o A136412 a136412 = (`div` 3) . (+ 1) . (* 5) . (4 ^)
%o A136412 -- _Reinhard Zumkeller_, Jun 17 2012
%o A136412 (PARI) a(n)=(5*4^n+1)/3 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A136412 (SageMath) [(5*4^n+1)/3 for n in range(31)] # _G. C. Greubel_, Jan 19 2023
%Y A136412 Sequences of the form (m*4^n + 1)/3: A007583 (m=2), this sequence (m=5),  A199210 (m=11), A199210 (m=11), A206373 (m=14).
%Y A136412 Cf. A007302, A140660, A181565, A199115.
%K A136412 nonn,easy
%O A136412 0,1
%A A136412 _Paul Curtz_, Mar 31 2008
%E A136412 Formula in definition and more terms from _R. J. Mathar_, Apr 04 2008