The Ulam sequence
is defined by ,
, with the general term for 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, ... (OEIS A002858). Here, the first term 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 representable in 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.
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. Math4, 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. 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, and A007300/M1328
in "The On-Line Encyclopedia of Integer Sequences."Wolfram,
S. A
New Kind of Science. Champaign, IL: Wolfram Media, p. 908,
2002.
is defined by 
,
, with the general term 
for 
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 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 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.
