1,2

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.

A permutation of the positive integers. Bisections are the lower and upper Wythoff sequences.

The consecutive pairs (1,2), (3,5), (4,7), (6,10),... are the much-studied Wythoff pairs, arising in connection with Wythoff's game.

MathWorld, Wythoff's game

Index entries for sequences that are permutations of the natural numbers

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

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

Cf. A072062, A000201, A001950, A094077, A026272.

Sequence in context: A072062 A002192 A095721 this_sequence A101212 A064706 A100282

Adjacent sequences: A072058 A072059 A072060 this_sequence A072062 A072063 A072064

nonn

Clark Kimberling (ck6(AT)evansville.edu), Jun 11 2002, Aug 17 2007

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 26 2008