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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. 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

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Last modified December 7 01:49 EST 2023. Contains 367616 sequences. (Running on oeis4.)