cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

R. L. Graham, May 02 2017

Keywords

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

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

A286281 a(n) = floor the elevator is on at the n-th stage of Ken Knowlton's elevator problem, version 2.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4
Offset: 1

Views

Author

N. J. A. Sloane, May 09 2017

Keywords

Comments

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 (except that stops when the elevator is going down don't count), otherwise down.

References

  • Ken Knowlton, Email to R. L. Graham and N. J. A. Sloane, May 04 2017

Links

Crossrefs

For records see A286282.
See A285200 for the first version of the elevator problem.

Programs

  • Maple
    hit:=Array(1..50, 0);
hit[1]:=1; a:=[1]; dir:=1; f:=1;
for s from 2 to 1000 do
if dir>0 or f=1 then f:=f+1; hit[f]:=hit[f]+1; dir:=1; else f:=f-1; dir:=-1; fi;
a:=[op(a), f];
if (dir=1) and ((hit[f] mod f) = 0) then dir:=1; else dir:=-1; fi;
od:
a;
  • Mathematica
    f[n_, m_: 20] := Block[{a = {}, r = ConstantArray[0, m], f = 1, d = 0}, Do[AppendTo[a, f]; If[d == 1, r = MapAt[# + 1 &, r, f]]; If[Or[And[ Divisible[r[[f]], f], d == 1], f == 1], f++; d = 1, f--; d = -1], {i, n}]; a]; f@ 100 (* Michael De Vlieger, May 10 2017 *)

A360830 Numbers that when concatenated with the natural numbers from 1 to N are divisible by the corresponding order number.

Original entry on oeis.org

1, 3, 6, 42, 84, 252, 2772, 36036, 612612, 11639628, 267711444, 803134332, 23290895628, 722017764468, 1444035528936, 53429314570632, 2190601897395912, 94195881588024216, 4427206434637138152, 30990445042459967064
Offset: 1

Views

Author

Rodolfo Kurchan, Feb 22 2023

Keywords

Comments

There is an elevator with the numbers from 1 to N. Each number goes up in the elevator and can go up as long as the concatenation of this number with the number of the floor (1 to N) is a multiple of the floor.
Example, the number 1 reaches floor 2, because 11 is divisible by 1, 12 is divisible by 2, but it does not reach floor 3 because 13 is not divisible by 3.
The sequence shows the numbers that can go higher in the elevator than the previous number.
For example, 2 cannot go higher than 1, so it does not appear, instead number 3 can go up to floor 3, since 31 is divisible by 1, 32 is divisible by 2 and 33 is divisible by 3, instead I couldn't get to floor 4 because 34 is not divisible by 4.

Examples

			42 goes after 6, because it is the smallest number that can go more than the 6th floor that can go number 6. 421, 422, 423, 424, 425, 426 and 427 are divisible by 1, 2, 3, 4, 5, 6 and 7, but 42 cannot go to floor 8th, because 428 it is not divisible by 8.
   n |         a(n)         | Maximum Elevator floor
  ---------------------------------------------------
   1 |                    1 |          2
   2 |                    3 |          3
   3 |                    6 |          6
   4 |                   42 |          7
   5 |                   84 |          8
   6 |                  252 |         10
   7 |                 2772 |         12
   8 |                36036 |         16
   9 |               612612 |         18
  10 |             11639628 |         22
  11 |            267711444 |         26
  12 |            803134332 |         28
  13 |          23290895628 |         30
  14 |         722017764468 |         31
  15 |        1444035528936 |         36
  16 |       53429314570632 |         40
  17 |     2190601897395912 |         42
  18 |    94195881588024216 |         46
  19 |  4427206434637138152 |         48
  20 | 30990445042459967064 |         52
  ...

References

  • Jaime Poniachik, Problem El Ascensor, La Odisea del Ingenio, May 1990.

Crossrefs

Cf. A285201, A286282.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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