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%I #107 Sep 30 2023 09:22:05
%S 0,0,0,2,2,2,2,4,4,6,6,10,10,16,16,26,26,42,42,68,68,110,110,178,178,
%T 288,288,466,466,754,754,1220,1220,1974,1974,3194,3194,5168,5168,8362,
%U 8362,13530,13530,21892,21892,35422,35422,57314,57314,92736,92736,150050,150050,242786,242786,392836,392836,635622,635622
%N Number of n-digit numbers N that do not end with 0 and are such that the reversal of N divides N but is different from N.
%C For the actual numbers, see A031877 and their reversals in A008919. See especially the comments in A008919.
%D W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays, Macmillan, New York, 1939, page 13; Dover, New York, 13th ed. 1987, pp. 14-15.
%D H. Camous, Jouer Avec Les Maths, "Cardinaux Réversibles", Section I, Problem 6, pp. 27, 37-38; Les Editions D'Organisation, Paris, 1984.
%D Heinrich Dörrie, Mathematische Miniaturen, Ferdinand Hirt, Breslau, Germany, 1943; see pages 337-339.
%D M. Gardner, Mathematical Magic Show, Vintage Books, 1978, pp. 203, 204, 211, 212.
%D C. A. Grimm and D. W. Ballew, Reversible multiples, J. Rec. Math. 8 (1975-1976), 89-91.
%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, London, 1986, Entry 1089.
%H Vincenzo Librandi, <a href="/A214927/b214927.txt">Table of n, a(n) for n = 1..1000</a>
%H Yuhong Guo, <a href="https://doi.org/10.3770/j.issn:2095-2651.2018.02.003">Some Identities for Palindromic Compositions Without 2's</a>, Journal of Mathematical Research with Applications 38.2 (2018): 130-136.
%H G. H. Hardy, <a href="http://www.math.ualberta.ca/mss/misc/A%20Mathematician's%20Apology.pdf">A Mathematician's Apology</a>, Cambridge Univ. Press, 1940, reprinted 2000, pp. 24-25.
%H J. Jonesco (proposer), E.-N. Barisien and [no initials given] Welsch (solvers), Problem 1622, L'Intermédiaire des mathématiciens, VI (1899), <a href="https://archive.org/details/lintermdiairede03lemogoog/page/200/mode/1up?view=theater">p. 200</a>; L'Intermédiaire des mathématiciens, XV (1908), <a href="https://archive.org/details/lintermdiairede00laisgoog/page/n143/mode/2up?view=theater">pp. 132-133</a>, <a href="https://archive.org/details/lintermdiairede00laisgoog/page/278/mode/2up?view=theater">pp. 278-279</a> (in French).
%H T. J. Kaczynski, <a href="http://www.jstor.org/stable/2689056">Note on a Problem of Alan Sutcliffe</a>, Math. Mag., 41 (1968), 84-86.
%H Leonard F. Klosinski and Dennis C. Smolarski, <a href="http://www.jstor.org/stable/2688542">On the Reversing of Digits</a>, Math. Mag., 42 (1969), 208-210.
%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/mm005281.pdf">Digit Reversal Without Apology</a>, Mathematics Magazine, Vol. 80 (2007), pp. 129-132. Also arXiv:math/0511366 [math.HO], 2005-2006.
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1307.0453">2178 And All That</a>, Fib. Quart., 52 (2014), 99-120.
%H Alan Sutcliffe, <a href="http://www.jstor.org/stable/2689015">Integers That Are Multiplied When Their Digits Are Reversed</a>, Mathematics Magazine, 39 (1966), 282-287.
%H Anne Ludington Young, <a href="https://www.fq.math.ca/Scanned/30-2/ludington1.pdf">k-Reverse multiples</a>, Fib. Q., 30 (1992), 126-132.
%H Anne Ludington Young, <a href="https://www.fq.math.ca/Scanned/30-2/ludington2.pdf">Trees for k-reverse multiples</a>, Fib. Q., 30 (1992), 166-174.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).
%F a(n) = 2*Fibonacci(floor((n-2)/2)) = 2*A103609(n-2).
%F G.f.: -2*x^4*(x+1) / (x^4+x^2-1). - _Colin Barker_, Dec 31 2013
%e The smallest examples of such numbers are 8712 and 9801 (so a(n)=0 for n < 4, a(4) = 2); 87912 and 98901 (so a(5) = 2); and 879912 and 989901 (so a(6) = 2).
%t Join[{0}, Table[2 Fibonacci[Floor[(n - 2) / 2]], {n, 2, 60}]] (* _Vincenzo Librandi_, Jun 18 2013 *)
%o (Magma) [0] cat [2*Fibonacci(Floor((n-2)/2)): n in [2..60]]; // _Vincenzo Librandi_, Jun 18 2013
%Y Cf. A000045, A001232, A008918, A008919, A031877, A103609, A222809, A222810, A222811, A222812.
%Y See also A222817, A222818, A222819, A222820.
%K nonn,base,easy
%O 1,4
%A _Gregory A. Rosenthal_, Mar 10 2013
%E Formula, more terms and additional references and links from _N. J. A. Sloane_, Mar 11 2013
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