A282442 a(n) is the smallest step size that does not occur on a staircase of n steps when following the following procedure: Take steps of length 1 up a staircase until you can't step any further, then take steps of length 2 down until you can't step any further, and so on.

2, 3, 3, 4, 6, 5, 5, 9, 9, 8, 10, 11, 11, 15, 15, 11, 12, 18, 19, 16, 20, 17, 15, 24, 25, 18, 20, 28, 19, 24, 26, 21, 21, 31, 31, 20, 28, 25, 21, 32, 40, 33, 31, 39, 39, 25, 25, 35, 35, 51, 47, 32, 40, 54, 55, 48, 50, 41, 39, 60, 59, 58, 63, 59, 49, 50, 58

Offset

1,1

Comments

a(n) <= n + 1.

From the Mathematics Stack Exchange question:

Assume there are n stairs (so n+1 places to stand).

Starting from the bottom, go up 1 stair at a time, until you reach the top;

then turn around and go down 2 stairs at a time, until you can't go further;

then turn around and go up 3 stairs at a time, until you can't go further;

then 4, 5, 6, etc. stairs at a time, until you can't even make one step.

Links

Peter Kagey, Table of n, a(n) for n = 1..10000

Sheljohn, A curious sequence, Mathematics Stack Exchange, Feb 15 2017.

Example

For n = 4:
step size 1: 0 -> 1 -> 2 -> 3 -> 4;
step size 2: 4 -> 2 -> 0;
step size 3: 0 -> 3.
Because the walker cannot take four steps down, a(4) = 4.

Maple

A282442 := proc(n)
    local h, dir, ss, ns;
    h := 0 ;
    dir := 1 ;
    for ss from 1 do
        if dir > 0 then
            ns := floor((n-h)/ss) ;
        else
            ns := floor(h/ss) ;
        end if;
        if ns = 0 then
            return ss;
        end if;
        h := h+dir*ns*ss ;
        dir := -dir ;
    end do:
end proc:
seq(A282442(n), n=1..100) ; # R. J. Mathar, Feb 25 2017

Mathematica

a[n_] := Module[{h = 0, dir = 1, ss, ns}, For[ss = 1, True, ss++, If[dir > 0, ns = Floor[(n - h)/ss], ns = Floor[h/ss]]; If[ns == 0, Return[ss]]; h = h + dir ns ss; dir = -dir]];
Array[a, 100] (* Jean-François Alcover, Mar 29 2020, after R. J. Mathar *)

Crossrefs

Sequence in context: A003977 A003971 A349371 * A200668 A200469 A200251

Adjacent sequences: A282439 A282440 A282441 * A282443 A282444 A282445

Keyword

nonn,look

Author

Peter Kagey, Feb 15 2017

Status

approved