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A006960 Reverse and Add! sequence starting with 196.

%I A006960 M5410 #108 Aug 17 2025 08:23:26
%S A006960 196,887,1675,7436,13783,52514,94039,187088,1067869,10755470,18211171,
%T A006960 35322452,60744805,111589511,227574622,454050344,897100798,1794102596,
%U A006960 8746117567,16403234045,70446464506,130992928913,450822227944,900544455998,1800098901007,8801197801088,17602285712176
%N A006960 Reverse and Add! sequence starting with 196.
%C A006960 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.
%C A006960 From _A.H.M. Smeets_, Jan 31 2019: (Start)
%C A006960 Palindromes for a(9)/2, a(14)/2 and a(20)/2.
%C A006960 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)
%C A006960 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
%D A006960 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 196, p. 58, Ellipses, Paris 2008.
%D A006960 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
%D A006960 Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.
%D A006960 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 70.
%D A006960 Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975), pages PC30-6 to PC30-9.
%D A006960 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006960 T. D. Noe and Michael Lee, <a href="/A006960/b006960.txt">Table of n, a(n) for n = 0..2390</a> (T. D. Noe supplied terms 0 to 200)
%H A006960 Patrick De Geest, <a href="http://www.worldofnumbers.com/weblinks.htm">Some thematic websources</a>
%H A006960 Jason Doucette, <a href="http://www.jasondoucette.com/worldrecords.html">World Records</a>
%H A006960 Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, <a href="http://arxiv.org/abs/1210.7593">On Polynomial Pairs of Integers</a>, arXiv:1210.7593 [math.NT], 2012. - From _N. J. A. Sloane_, Nov 08 2012
%H A006960 Felix Fröhlich, <a href="/A006960/a006960.txt">C++ program for this sequence</a>
%H A006960 Fred Gruenberger, <a href="https://www.jstor.org/stable/24969338">How to handle numbers with thousands of digits, and why one might want to</a>, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.
%H A006960 R. K. Guy, <a href="https://www.jstor.org/stable/25678158">What's left?</a>, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
%H A006960 Tim Irvin, <a href="http://www.fourmilab.ch/documents/threeyears/two_months_more.html">About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing</a>
%H A006960 Madras Math's Amazing Number Facts, <a href="https://web.archive.org/web/20081123022656/http://www.users.zetnet.co.uk/madras/maths/amazingnofacts/fact044.html">The Ultimate Palindrome</a>
%H A006960 I. Peter, <a href="https://web.archive.org/web/20110728141631/http://www.floot.demon.co.uk/palindromes.html">More trajectories</a>
%H A006960 Wade VanLandingham, <a href="https://web.archive.org/web/20061022000805/http://home.cfl.rr.com/p196/">196 and Other Lychrel Numbers</a>
%H A006960 John Walker, <a href="http://www.fourmilab.ch/documents/threeyears/threeyears.html">Three Years Of Computing: Final Report On The Palindrome Quest</a>
%H A006960 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/196-Algorithm.html">196-Algorithm.</a>
%H A006960 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PalindromicNumberConjecture.html">Palindromic Number Conjecture.</a>
%H A006960 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F A006960 a(n+1) = A056964(a(n)). - _A.H.M. Smeets_, Jan 27 2019
%e A006960 From _M. F. Hasler_, Apr 13 2019: (Start)
%e A006960 Start with 196 = a(0), then:
%e A006960 A056964(196) = 196 + 691 = 887 = a(1); then:
%e A006960 A056964(887) = 887 + 788 = 1675 = a(2); then:
%e A006960 A056964(1675) = 1675 + 5761 = 7436 = a(3); then:
%e A006960 A056964(7436) = 7436 + 6347 = 13783 = a(4); then:
%e A006960 A056964(13783) = 13783 + 38731 = 52514 = a(5); etc. (End)
%p A006960 a:= proc(n) option remember; `if`(n=0, 196, (h-> h+ (s->
%p A006960       parse(cat(s[-i]$i=1..length(s))))(""||h))(a(n-1)))
%p A006960     end:
%p A006960 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 25 2014
%t A006960 a = {196}; For[i = 2, i < 26, i++, a = Append[a, a[[i - 1]] + ToExpression[ StringReverse[ToString[a[[i - 1]]]]]]]; a
%t A006960 NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,196,25] (* _Harvey P. Dale_, Jun 05 2011 *)
%t A006960 NestList[#+IntegerReverse[#]&,196,25] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jan 04 2019 *)
%o A006960 (Haskell)
%o A006960 a006960 n = a006960_list !! n
%o A006960 a006960_list = iterate a056964 196 -- _Reinhard Zumkeller_, Sep 22 2011
%o A006960 (PARI) A006960_vec(N=99)=vector(N,i,N=if(i>1,A056964(N),196)) \\ _M. F. Hasler_, Apr 13 2019
%Y A006960 Cf. A023108, A023109, A033665, A016016, A056964, A004086.
%K A006960 nonn,base,nice,easy
%O A006960 0,1
%A A006960 _N. J. A. Sloane_, _Simon Plouffe_
%E A006960 More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002