A072061 - OEIS

%I #29 Feb 16 2025 08:32:46

%S 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 22,36,24,39,25,41,27,44,29,47,30,49,32,52,33,54,35,57,37,60,38,62,40,

%U 65,42,68,43,70,45,73,46,75,48,78,50,81,51,83,53,86,55,89,56,91,58,94

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

%C 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.

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

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

%C 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

%C 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

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 40.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WythoffsGame.html">Wythoff's game</a>.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F a(n) = n*(1+(-1)^n)/4+floor((2*n+1-(-1)^n)*(1+sqrt(5))/8). - _Wesley Ivan Hurt_, Apr 10 2015

%p 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

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

%o (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

%o (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

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

%K nonn,easy,changed

%O 1,2

%A _Clark Kimberling_, Jun 11 2002, Aug 17 2007

%E Edited by _N. J. A. Sloane_, Jul 26 2008