A214927 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.
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, 288, 288, 466, 466, 754, 754, 1220, 1220, 1974, 1974, 3194, 3194, 5168, 5168, 8362, 8362, 13530, 13530, 21892, 21892, 35422, 35422, 57314, 57314, 92736, 92736, 150050, 150050, 242786, 242786, 392836, 392836, 635622, 635622
Offset: 1
Examples
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).
References
- 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.
- H. Camous, Jouer Avec Les Maths, "Cardinaux Réversibles", Section I, Problem 6, pp. 27, 37-38; Les Editions D'Organisation, Paris, 1984.
- Heinrich Dörrie, Mathematische Miniaturen, Ferdinand Hirt, Breslau, Germany, 1943; see pages 337-339.
- M. Gardner, Mathematical Magic Show, Vintage Books, 1978, pp. 203, 204, 211, 212.
- C. A. Grimm and D. W. Ballew, Reversible multiples, J. Rec. Math. 8 (1975-1976), 89-91.
- D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, London, 1986, Entry 1089.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Yuhong Guo, Some Identities for Palindromic Compositions Without 2's, Journal of Mathematical Research with Applications 38.2 (2018): 130-136.
- G. H. Hardy, A Mathematician's Apology, Cambridge Univ. Press, 1940, reprinted 2000, pp. 24-25.
- J. Jonesco (proposer), E.-N. Barisien and [no initials given] Welsch (solvers), Problem 1622, L'Intermédiaire des mathématiciens, VI (1899), p. 200; L'Intermédiaire des mathématiciens, XV (1908), pp. 132-133, pp. 278-279 (in French).
- T. J. Kaczynski, Note on a Problem of Alan Sutcliffe, Math. Mag., 41 (1968), 84-86.
- Leonard F. Klosinski and Dennis C. Smolarski, On the Reversing of Digits, Math. Mag., 42 (1969), 208-210.
- Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132. Also arXiv:math/0511366 [math.HO], 2005-2006.
- N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
- N. J. A. Sloane, 2178 And All That [Local copy]
- Alan Sutcliffe, Integers That Are Multiplied When Their Digits Are Reversed, Mathematics Magazine, 39 (1966), 282-287.
- Anne Ludington Young, k-Reverse multiples, Fib. Q., 30 (1992), 126-132.
- Anne Ludington Young, Trees for k-reverse multiples, Fib. Q., 30 (1992), 166-174.
- Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).
Crossrefs
Programs
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Magma
[0] cat [2*Fibonacci(Floor((n-2)/2)): n in [2..60]]; // Vincenzo Librandi, Jun 18 2013
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Mathematica
Join[{0}, Table[2 Fibonacci[Floor[(n-2)/2]], {n, 2, 60}]] (* Vincenzo Librandi, Jun 18 2013 *) -
SageMath
def A214927(n): return 2*(fibonacci((n-2)//2) -int(n==1)) [A214927(n) for n in range(1,71)] # G. C. Greubel, Oct 23 2024
Formula
a(n) = 2*Fibonacci(floor((n-2)/2)) = 2*A103609(n-2), for n > 1.
G.f.: 2*x^4*(1+x) / (1-x^2-x^4). - Colin Barker, Dec 31 2013
Extensions
Formula, more terms and additional references and links from N. J. A. Sloane, Mar 11 2013
Comments