A285203 -id:A285203 - cp's OEIS frontend

A285200 a(n) = floor the elevator is on at the n-th stage of Ken Knowlton's elevator problem, version 1.

1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1

Offset: 1

N. J. A. Sloane, May 02 2017

nonn

An elevator steps up or down a floor at a time. It starts at floor 1, and always goes up from floor 1. From each floor m, it steps up every m-th time it stops there, otherwise down.

See A285202 for an alternative way to display this sequence.

Ken Knowlton, Email to R. L. Graham, Apr 26 2017

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Ken Knowlton, Illustration showing what floor the elevator is on for the first 49 stages

Cf. A285201, A285202, A285203.

See A286281 for a second version of the elevator problem.

hit:=Array(1..50,0);
hit[1]:=1; a:=[1]; dir:=1; f:=1;
for s from 2 to 1000 do
if dir>0 then f:=f+1; else f:=f-1; fi;
hit[f]:=hit[f]+1; a:=[op(a),f];
if (hit[f] mod f) = 0 then dir:=1; else dir:=-1; fi;
od:
a;

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

Original entry on oeis.org

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

R. L. Graham, May 02 2017

nonn

Indices of records in A285200.

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

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

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

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

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

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

A285202 A285200 displayed as an irregular triangle read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1

Offset: 0

N. J. A. Sloane, May 03 2017

This is Ken Knowlton's elevator sequence A285200, but with an extra 1 each time the elevator returns to (or starts at) the first floor.

Row lengths are A285204. The row maxima are given in A285203 (for n>0).

			Triangle begins:
1,
1, 2, 1,
1, 2, 3, 2, 1,
1, 2, 3, 2, 1,
1, 2, 3, 4, 3, 2, 1,
1, 2, 3, 2, 1,
1, 2, 3, 4, 3, 2, 1,
1, 2, 3, 2, 1,
1, 2, 3, 4, 3, 2, 1,
1, 2, 3, 2, 1,
1, 2, 3, 4, 5, 4, 3, 2, 1,
...

Cf. A285200, A285201, A285203, A285204.

A285204 Row lengths of triangle A285202.

Original entry on oeis.org

1, 3, 5, 5, 7, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 11, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 11, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 9, 5, 7, 5, 7, 5, 11, 5, 7, 5, 7, 5, 9, 5, 7

Offset: 0

N. J. A. Sloane, May 03 2017

nonn

Cf. A285200-A285203.

a(n) = 2*A285203(n)-1 for n>0.

cp's OEIS Frontend

A285200 a(n) = floor the elevator is on at the n-th stage of Ken Knowlton's elevator problem, version 1.

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

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A285202 A285200 displayed as an irregular triangle read by rows.

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A285204 Row lengths of triangle A285202.

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