An Ulam sequence with initial positive integers is the increasing
sequence
defined by
and . For , is the least integer greater
than
having exactly one representation as the sum with . 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, ... (OEIS A002858).
Here, the first term after the initial (1, 2) is 3 since . The next term is . The representation is excluded because its summands are not distinct. 5 is
not a member of the sequence since it is representable
in two ways, ,
but
is a member.
Proceeding in this manner, we can generate Ulam sequences for other initial ordered pairs,
examples of which are given in the table below.
Schmerl and Spiegel (1994) proved that Ulam sequences for odd 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 for (mod 4) have exactly three even
terms.
Cassaigne, J. and Finch, S. "A Class of 1-Additive Sequences and Quadratic Recurrences." Exper. Math.4, 49-60, 1995.
https://doi.org/10.1080/10586458.1995.10504307.Finch,
S. "Conjectures About 1-Additive Sequences." Fib. Quart.29,
209-214, 1991.Finch, S. "Are 0-Additive Sequences Always Regular?"
Amer. Math. Monthly99, 671-673, 1992a.Finch, S. "On
the Regularity of Certain 1-Additive Sequences." J. Combin. Th. Ser. A60,
123-130, 1992b.Finch, S. "Patterns in 1-Additive Sequences."
Exper. Math.1, 57-63, 1992c.Finch, S. R. "Stolarsky-Harborth
Constant." §2.16 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 145-151,
2003.Guy, R. K. "A Quarter Century of Monthly Unsolved
Problems, 1969-1993." Amer. Math. Monthly100, 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. Monthly102, 921-926,
1995.Recaman, B. "Questions on a Sequence of Ulam." Amer.
Math. Monthly80, 919-920, 1973.Schmerl, J. and Spiegel,
E. "The Regularity of Some 1-Additive Sequences." J. Combin. Theory
Ser. A66, 172-175, 1994.Sloane, N. J. A. Sequences
A001857/M0634, A002858/M0557,
A002859/M2303, A003666/M3237,
A003667/M3746, A007300/M1328,
and A048951 in "The On-Line Encyclopedia
of Integer Sequences."Wolfram, S. A
New Kind of Science. Champaign, IL: Wolfram Media, p. 908,
2002.
is the increasing
sequence 
defined by 
and 
. For 
, 
is the least integer greater
than 
having exactly one representation as the sum 
with 
. The numbers so
produced are sometimes called u-numbers or Ulam numbers.
. The next term is 
. The representation 
is excluded because its summands are not distinct. 5 is
not a member of the sequence since it is representable
in two ways, 
,
but 
is a member.
,
examples of which are given in the table below.
for odd 
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 
for 
(mod 4) have exactly three even
terms.
