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A258898 a(1)=a(2)=1, a(n) = ceiling(e*a(n-1) - a(n-2)) for n>2.

%I A258898 #11 Sep 08 2022 08:46:12
%S A258898 1,1,2,5,12,28,65,149,341,778,1774,4045,9222,21023,47925,109251,
%T A258898 249051,567740,1294227,2950334,6725613,15331778,34950481,79673480,
%U A258898 181624492,414033077,943834098,2151574001,4904750412,11180919918
%N A258898 a(1)=a(2)=1, a(n) = ceiling(e*a(n-1) - a(n-2)) for n>2.
%C A258898 Ratio of consecutive terms approaches A189040, (e + sqrt(e^2 - 4))/2.
%e A258898 a(2) = ceiling(e*1 - 1) = 2;
%e A258898 a(3) = ceiling(e*2 - 1) = 5;
%e A258898 a(4) = ceiling(e*5 - 2) = 12;
%e A258898 a(5) = ceiling(e*12 - 5) = 28.
%p A258898 a:= proc(n) option remember; `if`(n<3, 1,
%p A258898       ceil(exp(1)*a(n-1)-a(n-2)))
%p A258898     end:
%p A258898 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jun 18 2015
%t A258898 nxt[{a_,b_}]:={b,Ceiling[E*b-a]}; NestList[nxt,{1,1},30][[All,1]] (* _Harvey P. Dale_, Dec 02 2017 *)
%o A258898 (Magma) I:=[1,1]; [n le 2 select I[n] else Ceiling(Exp(1)*Self(n-1)-Self(n-2)): n in [1..200]];
%Y A258898 Cf. A001113, A189040.
%K A258898 nonn
%O A258898 1,3
%A A258898 _Morris Neene_, Jun 14 2015