A286496 Renyi-Ulam liar numbers: maximum k such that n questions "Is x in subset S of {1,...,k}?" are guaranteed to determine x when at most one answer can be a lie.

1, 1, 1, 2, 2, 4, 8, 16, 28, 50, 92, 170, 314, 584, 1092, 2048, 3854, 7280, 13796, 26214, 49932, 95324, 182360, 349524, 671088, 1290554, 2485512, 4793490, 9256394, 17895696, 34636832, 67108864, 130150524, 252645134, 490853404, 954437176, 1857283154, 3616814564

Offset

0,4

Comments

Calculated from Andrzej Pelc's complete solution for 1-liar games which gives the minimum number of questions required for a given set {1,...,k}.

Andrzej Pelc's formula for the minimum number of questions required for a given set {1,...,k} is: the least value of q such that f(q)>=k, where f(q)=2^q/(q+1) for even k, and f(q)=(2^q-q+1)/(q+1) for odd k. We use this to generate the sequence of maximum set sizes for each value of q.

Links

Table of n, a(n) for n=0..37.

D. Osthus and R. Watkinson, A simple solution to Ulam's liar game with one lie, Elemente der Mathematik 63 (2008), 97-101.

A. Pelc, Solution of Ulam’s Problem on searching with a lie, J. Combinatorial Theory, Series A, vol. 44 (1987), 129-140.

Example

a(1) = 1 since 1 question is (vacuously) sufficient to determine x in {1}; a(2) = 1, since 2 questions (with one possible lie) is no better than 1; a(3) >= 2, since we can determine x in {1,2} by asking "Is x in {1}?" three times and majority voting. But a(3) is not >2 because we need 5 questions for {1,2,3}; which implies a(4) = 2 also.

Maple

LiarSequence:=proc(n)
local q, L, k;
q:=1: L:=NULL;
for k from 1 to n do
   while 2^q/(q+1)-(k+1 mod 2)*(q-1)/(q+1)<k+1 do q:=q+1; L:=L, k end:
end: L end:

Crossrefs

Cf. A325908.

Sequence in context: A112433 A171648 A189914 * A318187 A217931 A090129

Adjacent sequences: A286493 A286494 A286495 * A286497 A286498 A286499

Keyword

nonn

Author

Robin W. Whitty, Jun 15 2017

Extensions

a(0) and a(29)-a(37) from Pontus von Brömssen, Jan 30 2024

Status

approved