A072061 [t], 1+[t], [2t], 2+[2t], [3t], 3+[3t], ..., where t=tau = (1+sqrt(5))/2 and []=floor.

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, 22, 36, 24, 39, 25, 41, 27, 44, 29, 47, 30, 49, 32, 52, 33, 54, 35, 57, 37, 60, 38, 62, 40, 65, 42, 68, 43, 70, 45, 73, 46, 75, 48, 78, 50, 81, 51, 83, 53, 86, 55, 89, 56, 91, 58, 94

Offset

1,2

Comments

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, 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.

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, Jul 08 2004

A more general conjecture may be stated as follows: 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, Jul 08 2004

Links

Table of n, a(n) for n=1..72.

MathWorld, Wythoff's game

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = n*(1+(-1)^n)/4+floor((2*n+1-(-1)^n)*(1+sqrt(5))/8). - Wesley Ivan Hurt, Apr 10 2015

Maple

A072061:=n->n*(1+(-1)^n)/4+floor((2*n+1-(-1)^n)*(1+sqrt(5))/8): seq(A072061(n), n=1..100); # Wesley Ivan Hurt, Apr 10 2015

Mathematica

Table[n*(1 + (-1)^n)/4 + Floor[(2 n + 1 - (-1)^n) (1 + Sqrt[5])/8], {n, 100}] (* Wesley Ivan Hurt, Apr 10 2015 *)

Prog

(Magma) [n*(1+(-1)^n)/4+Floor((2*n+1-(-1)^n)*(1+Sqrt(5))/8) : n in [1..100]]; // Wesley Ivan Hurt, Apr 10 2015
(PARI) lista(nn) = {v = []; for (n=1, nn, v = concat(v, nt = floor(n*(1+sqrt(5))/2)); v = concat(v, n+nt); ); v; } \\ Michel Marcus, Apr 14 2015

Crossrefs

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

Sequence in context: A265888 A340709 A095721 * A255557 A261102 A101212

Adjacent sequences: A072058 A072059 A072060 * A072062 A072063 A072064

Keyword

nonn,easy

Author

Clark Kimberling, Jun 11 2002, Aug 17 2007

Extensions

Edited by N. J. A. Sloane, Jul 26 2008

Status

approved