Self-Counting Sequence

The sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ... (OEIS A002024) consisting of 1 copy of 1, 2 copies of 2, 3 copies of 3, and so on. Surprisingly, there exist simple formulas for the th term ,

			(1)
			(2)

where  is the floor function and  is the ceiling function (Graham et al. 1994, p. 97). The sequence is also given by the recursive sequence

(3)

(Wolfram 2002, p. 129).

REFERENCES:

Gould, H. W. "Solution to Problem 571." Math. Mag. 38, 185-187, 1965.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Exercise 3.23 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 97, 1994.

Sloane, N. J. A. Sequence A002024/M0250 in "The On-Line Encyclopedia of Integer Sequences."Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 43, 1997.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 129, 2002.