Undulating Number
A number of the form 
, 
, etc. The first few nontrivial undulants (with the stipulation that 
)are 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, ... (Sloane's A046075). Including the trivial 1- and 2-digitundulants and dropping the requirement that gives Sloane's A033619.
The first few undulating A016073), withno larger such numbers of fewer than a million digits (Pickover 1995). Several tricks can be used to speed the search for squareundulating numbers, especially by examining the possible patterns of ending digits. For example, the only possible sets of fourtrailing digits for undulating Squares are 0404, 1616, 2121, 2929, 3636, 6161, 6464, 6969, 8484, and9696.
The only undulating Power 
for 
and up to 100 digits is 
(Pickover 1995).A large undulating prime is given by 
(Pickover 1995).
A binary undulant is a Power of 2 whose base-10 representation contains one or both of the sequences 
and 
. The first few are 
for 
, 107, 138, 159, 179, 187, 192, 199, 205, ... (Sloane's A046076). The smallest 
for which an undulating sequence of exactly 
-digit occurs for 
, 4, ... are , 138, 875, 949, 6617, 1802, 14545, ...(Sloane's A046077). An undulating binary sequence of length 10 occurs for 
(Pickover 1995).
References
Pickover, C. A. ``Is There a Double Smoothly Undulating Integer?'' In Computers, Pattern, Chaos and Beauty. New York: St. Martin's Press, 1990. Pickover, C. A. ``The Undulation of the Monks.'' Ch. 20 in Keys to Infinity. New York: W. H. Freeman, pp. 159-161 1995. Sloane, N. J. A. Sequences A016073,A033619,A046075,A046076, andA046077in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
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