Search: id:A072061
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%I A072061
%S A072061 1,2,3,5,4,7,6,10,8,13,9,15,11,18,12,20,14,23,16,26,17,28,19,31,21,34,
%T A072061 22,36,24,39,25,41,27,44,29,47,30,49,32,52,33,54,35,57,37,60,38,62,40,
%U A072061 65,42,68,43,70,45,73,46,75,48,78,50,81,51,83,53,86,55,89,56,91,58,94
%N A072061 [t], 1+[t], [2t], 2+[2t], [3t], 3+[3t], ..., where t=tau = (1+sqrt(5)/
               2 and []=floor.
%C A072061 The same sequence can be defined as follows: "a(1) = 1 and, for n>1, 
               a(n) = a(n-1) + n/2 if n is even, otherwise a(n) = smallest positive 
               integer which has not yet appeared in the sequence." This was originally 
               a separate entry in the database, contributed by John W. Layman (layman(AT)math.vt.edu), 
               Jul 08 2004. Antti Karttunen noticed on Jul 10 2004 that the two 
               entries appeared to be identical. This was finally proved by Clark 
               Kimberling, Aug 22 2007.
%C A072061 A permutation of the positive integers. Bisections are the lower and 
               upper Wythoff sequences.
%C A072061 The consecutive pairs (1,2), (3,5), (4,7), (6,10),... are the much-studied 
               Wythoff pairs, arising in connection with Wythoff's game.
%H A072061 MathWorld, <a href="http://mathworld.wolfram.com/WythoffsGame.html">Wythoff's 
               game</a>
%H A072061 <a href="Sindx_Per.html#IntegerPermutation">Index entries for sequences 
               that are permutations of the natural numbers</a>
%F A072061 Conjecture. For even n, the ratio a(n)/a(n-1) is asymptotic to (1 + sqrt(5))/
               2 as n becomes large. (At n=3000, the ratio is 1.61804697, compared 
               to the exact value 1.61803399.) - John W. Layman (layman(AT)math.vt.edu), 
               Jul 08 2004
%F A072061 A more general conjecture may be stated as follows. Conjecture. Define 
               {a(n)} by a(1)=1 and, for n>1, a(n)=a(n-1)+ Floor(kn) if n is even, 
               else a(n)=smallest positive integer which has not yet appeared in 
               the sequence, where k is a positive real number. Then a(2n)/a(2n-1) 
               is asymptotic to k+Sqrt(k^2+1) for large n. - John W. Layman (layman(AT)math.vt.edu), 
               Jul 08 2004
%Y A072061 Cf. A072062, A000201, A001950, A094077, A026272.
%Y A072061 Sequence in context: A072062 A002192 A095721 this_sequence A101212 A064706 
               A100282
%Y A072061 Adjacent sequences: A072058 A072059 A072060 this_sequence A072062 A072063 
               A072064
%K A072061 nonn
%O A072061 1,2
%A A072061 Clark Kimberling (ck6(AT)evansville.edu), Jun 11 2002, Aug 17 2007
%E A072061 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 26 2008

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