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A285201 Stage at which Ken Knowlton's elevator (version 1) reaches floor n for the first time.

%I A285201 #37 Nov 01 2020 06:08:02
%S A285201 1,2,5,14,45,174,825,4738,32137,251338,2224157,21952358,238962581,
%T A285201 2843085270,36696680241,510647009850,7619901954001,121367981060434,
%U A285201 2055085325869813,36861997532438654,698193329457246653,13924819967953406654,291683979376372766697,6402385486361598687666,146948520147021794869977
%N A285201 Stage at which Ken Knowlton's elevator (version 1) reaches floor n for the first time.
%C A285201 Indices of records in A285200.
%C A285201 When prefixed by a(0)=0, the first differences give A111063. - _N. J. A. Sloane_, May 03 2017
%H A285201 N. J. A. Sloane, <a href="/A285201/b285201.txt">Table of n, a(n) for n = 1..400</a>
%H A285201 Ken Knowlton, <a href="/A285200/a285200.png">Illustration showing what floor the elevator is on for the first 49 stages</a>
%F A285201 a(n) = 2 - n + 2 * Sum_{k=0..n-2} Sum_{j=0..k} k!/j!.
%F A285201 For n >= 2, a(n) = 1+n+2*Sum_{k=2..n} C(n,k)*(k-1)! = 1+n+2*n!*Sum_{k=2..n} 1/(k*(n-k)!). - _N. J. A. Sloane_, May 03 2017
%F A285201 E.g.f.: exp(x)*(1-x-2*log(1-x)). Omitting the factor exp(x), this gives (essentially) the e.g.f. for A098558 (or A052849). - _N. J. A. Sloane_, May 03 2017
%p A285201 a:= proc(n) option remember; `if`(n<3, n, ((n-1)^2*a(n-1)
%p A285201       -(n-2)*(2*n-3)*a(n-2)+(n-1)*(n-3)*a(n-3))/(n-2))
%p A285201     end:
%p A285201 seq(a(n), n=1..25);  # _Alois P. Heinz_, Jul 11 2018
%t A285201 a[n_] := 2 - n + 2 Sum[k!/j!, {k, 0, n-2}, {j, 0, k}];
%t A285201 Array[a, 25] (* _Jean-François Alcover_, Nov 01 2020 *)
%Y A285201 Cf. A285200, A285203, A111063, A098558, A052849, A286281, A286282.
%K A285201 nonn
%O A285201 1,2
%A A285201 R. L. Graham, May 02 2017