OFFSET
0,1
COMMENTS
196 is conjectured to be the smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.
From A.H.M. Smeets, Jan 31 2019: (Start)
Palindromes for a(9)/2, a(14)/2 and a(20)/2.
Observed: It seems that most, but not all, Lychrel numbers (seeds given in A063048) have a trajectory term that, divided by 2, becomes palindromic. Note that 196 is the first Lychrel number (A063048(1)). (End)
Observed: On average, 0.414 digits are gained by each step of the reverse and add procedure; i.e., 2.416 steps are needed on average to gain a factor of 10. This holds for any trajectory of reverse and add for decimal number representation. - A.H.M. Smeets, Feb 03 2019
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 196, p. 58, Ellipses, Paris 2008.
D. H. Lehmer, "Sujets d'étude. No. 74," Sphinx (Bruxelles), 8 (1938), 12-13. (This is the currently earliest known reference to the 196 Problem). - James D. Klein, Apr 09 2012.
Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 70.
Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975), pages PC30-6 to PC30-9.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe and Michael Lee, Table of n, a(n) for n = 0..2390 (T. D. Noe supplied terms 0 to 200)
Patrick De Geest, Some thematic websources
Jason Doucette, World Records
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Sole, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012. - From N. J. A. Sloane, Nov 08 2012
Felix Fröhlich, C++ program for this sequence
Fred Gruenberger, How to handle numbers with thousands of digits, and why one might want to, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.
R. K. Guy, What's left?, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
Madras Math's Amazing Number Facts, The Ultimate Palindrome
I. Peter, More trajectories
Wade VanLandingham, 196 and Other Lychrel Numbers
Eric Weisstein's World of Mathematics, 196-Algorithm.
Eric Weisstein's World of Mathematics, Palindromic Number Conjecture.
FORMULA
a(n+1) = A056964(a(n)). - A.H.M. Smeets, Jan 27 2019
EXAMPLE
From M. F. Hasler, Apr 13 2019: (Start)
Start with 196 = a(0), then:
A056964(196) = 196 + 691 = 887 = a(1); then:
A056964(887) = 887 + 788 = 1675 = a(2); then:
A056964(1675) = 1675 + 5761 = 7436 = a(3); then:
A056964(7436) = 7436 + 6347 = 13783 = a(4); then:
A056964(13783) = 13783 + 38731 = 52514 = a(5); etc. (End)
MAPLE
a:= proc(n) option remember; `if`(n=0, 196, (h-> h+ (s->
parse(cat(s[-i]$i=1..length(s))))(""||h))(a(n-1)))
end:
seq(a(n), n=0..40); # Alois P. Heinz, Jul 25 2014
MATHEMATICA
a = {196}; For[i = 2, i < 26, i++, a = Append[a, a[[i - 1]] + ToExpression[ StringReverse[ToString[a[[i - 1]]]]]]]; a
NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&, 196, 25] (* Harvey P. Dale, Jun 05 2011 *)
NestList[#+IntegerReverse[#]&, 196, 25] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2019 *)
PROG
(Haskell)
a006960 n = a006960_list !! n
a006960_list = iterate a056964 196 -- Reinhard Zumkeller, Sep 22 2011
CROSSREFS
KEYWORD
nonn,base,nice,easy
AUTHOR
EXTENSIONS
More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002
STATUS
approved