Ulam Sequence
The Ulam sequence 
is defined by 
, 
, with the general term 
for 
given by the leastInteger expressible uniquely as the Sum of two distinct earlier terms. The numbers so produced are sometimescalled 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 firstterm after the initial 1, 2 is obviously 3 since 
. The next term is 
. (We don't have to worry about 
since it is a sum of a single term instead of distinct terms.) 5 is not a member of the sequence since it is representablein two ways, 
, but 
is a member.
Proceeding in the manner, we can generate Ulam sequences for any 
, examples of which are given in the table below.
| 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 
for Odd 
have exactly two Even terms. Ulamsequences with only finitely many Even terms eventually must have periodic successive differences (Finch 1991, 1992abc).Cassaigne and Finch (1995) proved that the Ulam sequences 
for 
(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 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. SequenceA002858/M0557in ``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.
-Additive Sequences.'' http://www.mathsoft.com/asolve/sadd/sadd.html.
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