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Originally found at: http://203.162.7.73/webs/math/mathpages/www.seanet.com/~ksbrown/kmath312.htm
Self-Similar Reverse-Sum Sequences
For any positive integers N and B let R(N,B) denote the integer
given by reversing the digits of N in the base B. For example,
R(2316,10)=6132. It's interesting that for some initial integers
x_0 and bases B, the recurrence x_(k+1) = x_k + R(x_k,B) (1)produces a sequence that is "periodic" in the sense that self-similar
strings recur at regular intervals. For example, in the base 4
beginning with the number 255 (decimal), the recurrence (1) leads to
a palindrome-free sequence with the following six-step cycle 22 [n*0] 131 [n*3] 12
10 [(n+2)*3] 23 [(n+2)*0]
11 [n*0] 3222 [n*3] 01
22 [n*0] 2111 [n*3] 12
10 [(n+2)*3] 23 [(n+3)*0]
11 [(n+1)*0] 312 [(n+1)*3] 01
22 [(n+1)*0] 131 [(n+1)*3] 12where [n*j] denotes a string of n consecutive j's. Cycles of this
kind have been constructed for any base equal to a power of 2, and
for certain other bases. (See Dave Seal's results.)However, apparently no one knows of any such "cycle" in the base 10.
Of course, if we allow a doubly infinite string of digits then we
have the following self-similar sequences in the base 10 period=1 period=2 period=6 <..9999..> <..3333..> <..1111..>
<..9999..> <..6666..> <..2222..>
<..9999..> <..3333..> <..4444..>
<..9999..> <..6666..> <..8888..>
<..9999..> <..3333..> <..7777..>
<..9999..> <..6666..> <..5555..>
<..9999..> <..3333..> <..1111..>
etc. etc. etc.For finite strings of digits there are some sequences that maintain
their patterns for a long time. For example, consider the base-10
reverse-sum sequence beginning with the integer 17509097067. The first
16 values are
175 09097 067
935 88187 638
177 266376 177
948 940038 948
179 8770088 797
977 7570867 768
184 55251625 547
930 07866881 028 175 026733751 067
935 184071371 638
177 1357241853 177
948 4938669384 948
179 79778337779 797
977 77551725577 768
184 555104441155 547
930 106248842711 028 etcThe three leading and trailing digits of this sequence repeat every
8 steps, and this pattern continues for nearly 12 complete cycles
before it finally unravels. However, the interior digits don't seem
to exhibit any consistent pattern...except for the fact that they
support the 8-step cycle of the outer digits.If this were a 6-step cycle instead of an 8-step cycle it might be
possible to combine it with the 6-step doubly infinite pattern. Even
for the 8-step cycle of outer digits this come tantalizingly close to
working, as illustrated by the sequence below 935 1111111111 638
177 12222222223 177
948 44444444444 948
179 788888888889 797
977 777777777777 768
184 5555555555555 547
930 1111111111111 028
175 02222222222222 067
935 24444444444442 638The pattern of borrows and carries necessary to support the inner digit
cycle is nicely supported by the exterior digits, so [k*1] becomes
[(k+3)*1] after 6 steps. Unfortunately the pattern unravels by the
time the outer digits start to repeat.QUESTION: Does anyone know of any self-similar periodic reverse-sum
sequences in base 10?I suppose this sort of begs the question of what qualifies as a
self-similar periodic sequence. In general, the sequence of
integers N_k, k=0,1,2,... is self-similar with period p iff for
each j, j=0,1,..,p-1 every one of the numbers N_(np+j) is of the
same form
[E1(n)][E2(n)][E3(n)]...[Et(n)]where Ei(n) is a string of digits, possibly a function of n. For
example, in the base four we saw above that every term N_(6n+0) of
a particular sequence was of the form [22][n*0][131][n*3][12]so E1(n) = {22} for all n, whereas E2(n) = {n 0's}. We could allow
the E functions to be more complicated functions of n, injstead of
just determining the number of repetitions of a digit in a string,
but I've never seen a self-similar sequence with any other kind of
functionality.It's also interesting to observe the "palindromic quality" of the
terms of sequences produced by iterating (1). Notice that about
half of the iterates are such that every digit is within 1 unit of
its reflected counterpart. Actually, the study of these palindromic
effects are what has prompted most of the activity in this area, but
I think the general question of self-similar periodic sequences is
much more interesting.