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A174541 Baron Munchhausen's Sequence. 1
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 (list; graph; refs; listen; history; text; internal format)
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. Kahovanova, 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

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Last modified August 3 21:09 EDT 2021. Contains 346441 sequences. (Running on oeis4.)