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A206373 a(n) = (14*4^n + 1)/3.

%I A206373 #24 Jan 20 2023 01:33:24
%S A206373 5,19,75,299,1195,4779,19115,76459,305835,1223339,4893355,19573419,
%T A206373 78293675,313174699,1252698795,5010795179,20043180715,80172722859,
%U A206373 320690891435,1282763565739,5131054262955,20524217051819,82096868207275,328387472829099,1313549891316395
%N A206373 a(n) = (14*4^n + 1)/3.
%C A206373 A generalized Engel expansion of 2/7 to the base b := 4/3 as defined in A181565 with associated series expansion 2/7 = b/5 + b^2/(5*19) + b^3/(5*19*75) + b^4/(5*19*75*299) + .... - _Peter Bala_, Oct 30 2013
%H A206373 G. C. Greubel, <a href="/A206373/b206373.txt">Table of n, a(n) for n = 0..1000</a>
%H A206373 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F A206373 a(n) = (14*4^n + 1)/3.
%F A206373 From _Peter Bala_, Oct 30 2013: (Start)
%F A206373 a(n+1) = 4*a(n) - 1 with a(0) = 5.
%F A206373 a(n) = 5*a(n-1) - 4*a(n-2) with a(0) = 5 and a(1) = 19.
%F A206373 O.g.f. (5 - 6*x)/((1 - x)*(1 - 4*x)). (End)
%F A206373 E.g.f.: (1/3)*(14*exp(4*x) + exp(x)). - _G. C. Greubel_, Jan 19 2023
%t A206373 (14*4^Range[0,30]+1)/3 (* or *) LinearRecurrence[{5,-4},{5,19},30] (* _Harvey P. Dale_, Jan 13 2023 *)
%o A206373 (Magma) [(14*4^n+1)/3 : n in [0..30]];
%o A206373 (PARI) a(n)=(14*4^n + 1)/3 \\ _Charles R Greathouse IV_, Jun 01 2015
%o A206373 (SageMath) [(7*2^(2*n+1)+1)/3 for n in range(31)] # _G. C. Greubel_, Jan 19 2023
%Y A206373 Sequences of the form (m*4^n + 1)/3: A007583 (m=2), A136412 (m=5),  A199210 (m=11), A199210 (m=11), this sequence (m=14).
%Y A206373 Cf. A181565.
%K A206373 nonn,easy
%O A206373 0,1
%A A206373 _Brad Clardy_, Feb 07 2012