A286282 Stage at which Ken Knowlton's elevator (version 2) reaches floor n for the first time.

1, 2, 5, 18, 79, 408, 2469, 17314, 138555, 1247052, 12470593, 137176614, 1646119479, 21399553360, 299593747197, 4493906208138, 71902499330419, 1222342488617364, 22002164795112825, 418041131107143982, 8360822622142879983, 175577275065000480024, 3862700051430010560949

Offset

1,2

Comments

Indices of records in A286281.

Theorem: Let b(n) = Sum_{k=0..n} n!/k! = A000522(n). Then a(n) = 2*b(n-1)-n+2-2*(n-1)!. - R. L. Graham, May 10 2017

This implies the following recurrence (conjectured by N. J. A. Sloane on May 09 2017): a(1)=1, and for n>=1, a(n+1) = n*a(n) + n^2 - 3*n + 3. From the asymptotic expansion of b(n) (see A000522), we have a(n) ~ 2*(e-1)*(n-1)!.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..400

Formula

a(n) = 2*A002627(n-1) - (n-2). - N. J. A. Sloane, May 15 2017

Conjecture: a(n) +(-n-2)*a(n-1) +3*(n-1)*a(n-2) +(-3*n+8)*a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, May 21 2017

Conjecture: (n+1)*a(n) +(-n^2+3*n-27)*a(n-1) +3*(-n^2+10*n-13)*a(n-2) +(n-3)*(4*n-17)*a(n-3)=0. - R. J. Mathar, May 21 2017

Maple

A286282 := proc(n)
    2*A002627(n-1)-n+2 ;
end proc:
seq(A286282(n), n=1..21) ; # R. J. Mathar, May 21 2017

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]; Rest@ Map[First, Values@ PositionIndex@ FoldList[Max, 0, f@ 200000]] - 1 (* Michael De Vlieger, May 10 2017, Version 10 *)

Prog

(Python)
times = {1: 1, 2: 1, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 1, 9: 1, 10: 1, 11: 1, 12: 1, 13: 1, 14: 1, 15: 1, 16: 1}
first = {1: 0, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0, 10: 0, 11: 0, 12: 0, 13: 0, 14: 0, 15: 0, 16: 0}
floor = 1
steps = 1
while floor < 17:
....if first[floor] == 0:
........first[floor] = 1
........print("First Time: ", floor, steps)
....if floor == 1:
........floor += 1
....else:
........if times[floor] < floor:
............times[floor] += 1
............floor -= 1
........else:
............times[floor] = 0
............floor += 1
....steps += 1
print(floor, steps)
# David Consiglio, Jr., May 09 2017

Crossrefs

Cf. A000522, A002627, A286281, A285201.

Sequence in context: A338179 A206293 A137861 * A192637 A111916 A328440

Adjacent sequences: A286279 A286280 A286281 * A286283 A286284 A286285

Keyword

nonn

Author

N. J. A. Sloane, May 09 2017

Extensions

a(10)-a(13) from David Consiglio, Jr., May 09 2017

Further terms added by N. J. A. Sloane, May 10 2017 based on R. L. Graham's formula.

Status

approved