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A052968 a(n) = 1 + 2^(n-1) + n for n > 0, a(0) = 2.

%I A052968 #38 Jul 02 2025 16:01:58
%S A052968 2,3,5,8,13,22,39,72,137,266,523,1036,2061,4110,8207,16400,32785,
%T A052968 65554,131091,262164,524309,1048598,2097175,4194328,8388633,16777242,
%U A052968 33554459,67108892,134217757,268435486,536870943,1073741856,2147483681
%N A052968 a(n) = 1 + 2^(n-1) + n for n > 0, a(0) = 2.
%C A052968 O. M. Cain proves that the number of consecutive zeros in the decimal expansion of 5^(n+2^n+2), that is 5^a(n+1), is nondecreasing and grows to infinity as m gets bigger. See link. - _Michel Marcus_, Nov 07 2019
%H A052968 Vincenzo Librandi, <a href="/A052968/b052968.txt">Table of n, a(n) for n = 0..1000</a>
%H A052968 O. M. Cain, <a href="https://arxiv.org/abs/1910.13829">The Exceptional Selfcondensability of Powers of Five</a>, arXiv:1910.13829 [math.HO], 2019. See Theorem 10.4. p. 13.
%H A052968 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1040">Encyclopedia of Combinatorial Structures 1040</a>
%H A052968 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A052968 G.f.: (-2 + 5*x - 3*x^2 + x^3)/(-1+2*x)/(-1+x)^2
%F A052968 Recurrence: {a(3)=8, a(2)=5, a(1)=3, a(0)=2, 2*a(n)-a(n+1)-n=0}.
%p A052968 spec := [S,{S=Union(Sequence(Prod(Sequence(Z),Z)),Prod(Sequence(Z),Sequence(Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%t A052968 CoefficientList[Series[(-2+5*x-3*x^2+x^3)/(-1+2*x)/(-1+x)^2,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 22 2012 *)
%t A052968 Join[{2},Table[1+2^(n-1)+n,{n,40}]] (* or *) LinearRecurrence[{4,-5,2},{2,3,5,8},40] (* _Harvey P. Dale_, Feb 10 2018 *)
%o A052968 (Magma) I:=[2, 3, 5, 8]; [n le 4 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 22 2012
%K A052968 easy,nonn
%O A052968 0,1
%A A052968 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052968 More terms from _James Sellers_, Jun 05 2000