The Connell sequence is the sequence obtained by starting with the first positive odd number (1), taking the next two even numbers (2, 4), the next three odd numbers
(5, 7, 9), the next four even numbers (10, 12, 14, 16), and so on. The first few
terms are 1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, ... (OEIS A001614).
A binary plot of the sequence from 1 to 255 is illustrated
above.

Amazingly, the terms of this sequence have the closed form

Connell, I. "Elementary Problem E1382." Amer. Math. Monthly66, 724, 1959.Connell, I. "An Unusual
Sequence." Amer. Math. Monthly67, 380, 1960.Iannucci,
D. E. and Mills-Taylor, D. "On Generalizing the Connell Sequence."
J. Integer Sequences2, No. 99.1.7, 1999. http://www.cs.uwaterloo.ca/journals/JIS/.Lakhtakia,
A. and Pickover, C. "The Connell Sequence." J. Recr. Math.25,
90-92, 1993.Pickover, C. A. Computers
and the Imagination. New York: St. Martin's Press, p. 276, 1991.Pickover,
C. A. "Bird, Dog Dog, Bird, Bird Bird, Dog Dog Dog Dog." Ch. 39
in The
Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge
University Press, pp. 88-89 and 294-295, 2002.Sloane, N. J. A.
Sequence A001614/M0962 in "The On-Line
Encyclopedia of Integer Sequences."Stevens, G. E. "A
Connell-Like Sequence." J. Integer Sequences1, Np. 98.1.4,
1998. http://www.cs.uwaterloo.ca/journals/JIS/.