A006960 Reverse and Add! sequence starting with 196.

196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007, 8801197801088, 17602285712176

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• base
• nice
• easy

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

Examples

			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)
		

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 Solé, 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.
• Tim Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing
• Madras Math's Amazing Number Facts, The Ultimate Palindrome
• I. Peter, More trajectories
• Wade VanLandingham, 196 and Other Lychrel Numbers
• John Walker, Three Years Of Computing: Final Report On The Palindrome Quest
• Eric Weisstein's World of Mathematics, 196-Algorithm.
• Eric Weisstein's World of Mathematics, Palindromic Number Conjecture.
• Index entries for sequences related to Reverse and Add!

Crossrefs

Cf. A023108, A023109, A033665, A016016, A056964, A004086.

Programs

Haskell

a006960 n = a006960_list !! n
a006960_list = iterate a056964 196 -- Reinhard Zumkeller, Sep 22 2011

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 *)

PARI

A006960_vec(N=99)=vector(N,i,N=if(i>1,A056964(N),196)) \\ M. F. Hasler, Apr 13 2019

Formula

a(n+1) = A056964(a(n)). - A.H.M. Smeets, Jan 27 2019

Extensions

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002