A174541 Baron Munchhausen's Sequence.

0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset

1,5

Comments

Let n coins weighing 1, 2, ..., n grams be given. Suppose Baron Munchhausen knows which coin weighs how much, but his audience does not. Then a(n) is the minimum number of weighings the Baron must conduct on a balance scale, so as to unequivocally demonstrate the weight of at least one of the coins.

After a(1) = 0, a(n) is either 1 or 2 for all n.

a(n) = 1 for n triangular, n triangular-plus-one, T_n a square, and T_n a square-plus-one, where T_n is the n-th triangular number; a(n) = 2 for all other n > 1.

Links

Table of n, a(n) for n=1..105.

M. Brand, Tightening the bounds on the Baron's Omni-sequence, Discrete Math., 312 (2012), 1326-1335.

T. Khovanova, K. Knop and A. Radul, Baron Munchhausen's Sequence, arXiv:1003.3406 [math.CO], 2010.

T. Khovanova, K. Knop, A. Radul, Baron Munchhausen's Sequence, J. Int. Seq. 13 (2010) # 10.8.7.

T. Khovanova, A. Radul, Another Coins Sequence

Example

a(7) = 1 because the weighing 1 + 2 + 3 < 7 conclusively demonstrates the weight of the seven-gram coin.

Mathematica

triangularQ[n_] := IntegerQ[ Sqrt[8n+1]]; a[1] = 0; a[n_ /; triangularQ[n] || triangularQ[n-1] || IntegerQ[ Sqrt[n*(n+1)/2]] || IntegerQ[ Sqrt[n*(n+1)/2 - 1]]] = 1; a[_] = 2; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jul 30 2012, after comments *)

Prog

(Scheme) ;;; The following Scheme program generates terms of Baron
;;; Munchhausen's Sequence.
(define (acceptable? n)
..(or (triangle? n)
......(= n 2)
......(triangle? (- n 1))
......(square? (triangle n))
......(square? (- (triangle n) 1))))
(stream-map
.(lambda (n)
...(if (= n 1)
.......0
.......(if (acceptable? n)
...........1
...........2)))
.(the-integers))

Crossrefs

Cf. A000217, A000124, A001108, A072221, A186313.

Sequence in context: A070106 A182595 A109706 * A029444 A122191 A097847

Adjacent sequences: A174538 A174539 A174540 * A174542 A174543 A174544

Keyword

nonn,nice

Author

Tanya Khovanova, Konstantin Knop, and Alexey Radul, Mar 21 2010

Status

approved