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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. 30
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

For the actual numbers, see A031877 and their reversals in A008919. See especially the comments in A008919.

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 Reversibles", 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.

J. Jonesco (proposer), E.-N. Barisien and [no initials given] Welsch (solvers), Problem 1622, L'Intermédiaire des mathématiciens, XV (1908), 132-133, 278-279.

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.

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:0511366, 2005.

N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.

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).

FORMULA

a(n) = 2*Fibonacci(floor((n-2)/2)).

Empirical g.f.: -2*x^4*(x+1) / (x^4+x^2-1). - Colin Barker, Dec 31 2013

EXAMPLE

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).

MATHEMATICA

Join[{0}, Table[2 Fibonacci[Floor[(n - 2) / 2]], {n, 2, 60}]] (* Vincenzo Librandi, Jun 18 2013 *)

PROG

(MAGMA) [0] cat [2*Fibonacci(Floor((n-2)/2)): n in [2..60]]; // Vincenzo Librandi, Jun 18 2013

CROSSREFS

Cf. A000045, A001232, A008918, A008919, A031877, A103609, A222809, A222810, A222811, A222812.

See also A222817-A222820.

Sequence in context: A239494 A071809 A104976 * A239485 A237756 A010334

Adjacent sequences:  A214924 A214925 A214926 * A214928 A214929 A214930

KEYWORD

nonn,base

AUTHOR

Gregory A. Rosenthal, Mar 10 2013

EXTENSIONS

Formula, more terms and additional references and links from N. J. A. Sloane, Mar 11 2013

STATUS

approved

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Last modified January 22 09:52 EST 2019. Contains 319363 sequences. (Running on oeis4.)