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

1, 2, 5, 14, 45, 174, 825, 4738, 32137, 251338, 2224157, 21952358, 238962581, 2843085270, 36696680241, 510647009850, 7619901954001, 121367981060434, 2055085325869813, 36861997532438654, 698193329457246653, 13924819967953406654, 291683979376372766697, 6402385486361598687666, 146948520147021794869977

Offset: 1

Author

R. L. Graham, May 02 2017

Keywords

• nonn

Comments

Indices of records in A285200.

When prefixed by a(0)=0, the first differences give A111063. - N. J. A. Sloane, May 03 2017

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..400
• Ken Knowlton, Illustration showing what floor the elevator is on for the first 49 stages

Crossrefs

Cf. A285200, A285203, A111063, A098558, A052849, A286281, A286282.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, n, ((n-1)^2*a(n-1)
      -(n-2)*(2*n-3)*a(n-2)+(n-1)*(n-3)*a(n-3))/(n-2))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 11 2018

Mathematica

a[n_] := 2 - n + 2 Sum[k!/j!, {k, 0, n-2}, {j, 0, k}];
Array[a, 25] (* Jean-François Alcover, Nov 01 2020 *)

Formula

a(n) = 2 - n + 2 * Sum_{k=0..n-2} Sum_{j=0..k} k!/j!.

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

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