Ulam Sequence

The Ulam sequence $\{a_i\}=(u,v)$ is defined by $a_1=u$, $a_2=v$, with the general term $a_n$ for $n>2$ given by the least Integer expressible uniquely as the Sum of two distinct earlier terms. The numbers so produced are sometimes called U-Numbers or Ulam Numbers.

The first few numbers in the (1, 2) Ulam sequence are 1, 2, 3, 4, 6, 8, 11, 13, 16, ... (Sloane's A002858). Here, the first term after the initial 1, 2 is obviously 3 since $3=1+2$. The next term is $4=1+3$. (We don't have to worry about $4=2+2$ since it is a sum of a single term instead of distinct terms.) 5 is not a member of the sequence since it is representable in two ways, $5=1+4=2+3$, but $6=2+4$ is a member.

Proceeding in the manner, we can generate Ulam sequences for any $(u, v)$, examples of which are given in the table below.

$(u, v)$	Sloane	Sequence
(1, 2)	Sloane's A002858	1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...
(1, 3)	Sloane's A002859	1, 3, 4, 5, 6, 8, 10, 12, 17, 21, ...
(1, 4)	Sloane's A003666	1, 4, 5, 6, 7, 8, 10, 16, 18, 19, ...
(1, 5)	Sloane's A003667	1, 5, 6, 7, 8, 9, 10, 12, 20, 22, ...
(2, 3)	Sloane's A001857	2, 3, 5, 7, 8, 9, 13, 14, 18, 19, ...
(2, 4)	Sloane's A048951	2, 4, 6, 8, 12, 16, 22, 26, 32, 36, ...
(2, 5)	Sloane's A007300	2, 5, 7, 9, 11, 12, 13, 15, 19, 23, ...

Schmerl and Spiegel (1994) proved that Ulam sequences $(2,v)$ for Odd $v\geq 5$ have exactly two Even terms. Ulam sequences with only finitely many Even terms eventually must have periodic successive differences (Finch 1991, 1992abc). Cassaigne and Finch (1995) proved that the Ulam sequences $(4,v)$ for $5\leq v\equiv 1$ (mod 4) have exactly three Even terms.

The Ulam sequence can be generalized by the s-Additive Sequence.

See also Greedy Algorithm, s-Additive Sequence, Stöhr Sequence

References

Cassaigne, J. and Finch, S. ``A Class of 1-Additive Sequences and Quadratic Recurrences.'' Exper. Math 4, 49-60, 1995.

Finch, S. ``Conjectures About 1-Additive Sequences.'' Fib. Quart. 29, 209-214, 1991.

Finch, S. ``Are 0-Additive Sequences Always Regular?'' Amer. Math. Monthly 99, 671-673, 1992a.

Finch, S. ``On the Regularity of Certain 1-Additive Sequences.'' J. Combin. Th. Ser. A 60, 123-130, 1992b.

Finch, S. ``Patterns in 1-Additive Sequences.'' Exper. Math. 1, 57-63, 1992c.

Finch, S. ``Ulam $s$-Additive Sequences.'' http://www.mathsoft.com/asolve/sadd/sadd.html.

Guy, R. K. ``A Quarter Century of Monthly Unsolved Problems, 1969-1993.'' Amer. Math. Monthly 100, 945-949, 1993.

Guy, R. K. ``Ulam Numbers.'' §C4 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 109-110, 1994.

Guy, R. K. and Nowakowski, R. J. ``Monthly Unsolved Problems, 1969-1995.'' Amer. Math. Monthly 102, 921-926, 1995.

Recaman, B. ``Questions on a Sequence of Ulam.'' Amer. Math. Monthly 80, 919-920, 1973.

Schmerl, J. and Spiegel, E. ``The Regularity of Some 1-Additive Sequences.'' J. Combin. Theory Ser. A 66, 172-175, 1994.

Sloane, N. J. A. Sequence A002858/M0557 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.