A004278 1, 3 and the positive even numbers.

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

a(n) is the maximum number of turns that player A needs to identify with certainty the location of the coin that player B has hidden in one box in a row of n + 1 boxes. Player A starts by opening one of the boxes to see if the coin is in that box. After that, B secretly relocates the coin from its current box to one of the neighboring boxes, except when n = 1. In that case the game ends before B can relocate the coin. On each turn player A opens one box and when player A can tell in which box the coin is located, the game ends. Can be proved. - Bob Andriesse, Dec 22 2017

Links

• Iain Fox, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (2,-1)

Crossrefs

Cf. A005843.

Programs

Haskell

a004278 n = if n <= 3 then n else 2 * (n - 2)
a004278_list = [1, 2, 3] ++ [4, 6 ..]
-- Reinhard Zumkeller, Nov 10 2012

Mathematica

Sort[Join[{1,3},2*Range[60]]] (* Harvey P. Dale, Feb 04 2015 *)

PARI

first(n) = Vec((x + x^5)/(x - 1)^2 + O(x^(n+1))) \\ Iain Fox, Dec 21 2017

Formula

From Iain Fox, Dec 21 2017: (Start)

G.f.: (x + x^5)/(1 - x)^2.

a(n) = 2*a(n-1) - a(n-2), n > 2 and n is not 5.

a(n) = 2*n - 2 + floor((4/Pi)*arctan(2 - n)).

(End)

E.g.f.: 2*e^x*(x - 1) + 3 + 2*x + x^2/2. - Iain Fox, Dec 22 2017

a(n) = (abs(4 - n) + 3*n - 4)/2. - Iain Fox, Dec 23 2017

Extensions

Offset corrected by Reinhard Zumkeller, Nov 10 2012