“Ulam Sequence”的概念、定义、习题解题方法及翻译-数学辞典

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.

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. ``Ulam

-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. 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.