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

Author

Peter Kagey, Feb 15 2017

Keywords

• nonn
• look

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.

Examples

			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.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Sheljohn, A curious sequence, Mathematics Stack Exchange, Feb 15 2017.

Programs

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 *)