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A072061
[t], 1+[t], [2t], 2+[2t], [3t], 3+[3t], ..., where t=tau = (1+sqrt(5))/2 and []=floor.
5
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
REFERENCES
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 40.
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 11 2002, Aug 17 2007
EXTENSIONS
Edited by N. J. A. Sloane, Jul 26 2008
STATUS
approved