A001232 -id:A001232 - cp's OEIS frontend

A039684 Erroneous version of A001232.

1089, 10989, 109989, 1099989, 10999989, 109999989

Offset: 1

dead

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

Gregory A. Rosenthal, Mar 10 2013

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

			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).

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.

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).

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

See also A222817, A222818, A222819, A222820.

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

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

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

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

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

A103609 Fibonacci numbers repeated (cf. A000045).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 8, 13, 13, 21, 21, 34, 34, 55, 55, 89, 89, 144, 144, 233, 233, 377, 377, 610, 610, 987, 987, 1597, 1597, 2584, 2584, 4181, 4181, 6765, 6765, 10946, 10946, 17711, 17711, 28657, 28657, 46368, 46368, 75025, 75025, 121393

Offset: 0

Roger L. Bagula, Mar 24 2005

The usual policy in the OEIS is not to include such "doubled" sequences. This is an exception. - N. J. A. Sloane

The Gi2 sums, see A180662, of triangle A065941 equal the terms of this sequence without the two leading zeros. - Johannes W. Meijer, Aug 16 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]
I. Wloch, U. Bednarz, D. Bród, A Wloch and M. Wolowiec-Musial, On a new type of distance Fibonacci numbers, Discrete Applied Math., Volume 161, Issues 16-17, November 2013, Pages 2695-2701.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Partial sums: A094707.

Cf. A000045, A065941, A180662, A214927.

[Fibonacci(Floor(n/2)): n in [0..60]]; // G. C. Greubel, Oct 22 2024

A103609 := proc(n): combinat[fibonacci](floor(n/2)) ; end proc: seq(A103609(n), n=0..52); # Johannes W. Meijer, Aug 16 2011

a[0] = 0; a[1] = 0; a[2] = 1; a[3] = 1; a[n_Integer?Positive] := a[n] = a[n - 2] + a[n - 4]; aa = Table[a[n], {n, 0, 200}]
Join[{0, 0}, LinearRecurrence[{0, 1, 0, 1}, {1, 1, 1, 1}, 60]] (* Vincenzo Librandi, Jan 19 2016 *)
With[{fibs=Fibonacci[Range[0,30]]},Riffle[fibs,fibs]] (* Harvey P. Dale, Jul 11 2025 *)

a(n)=fibonacci(n\2) \\ Charles R Greathouse IV, Oct 07 2015

my(x='x+O('x^50)); Vec(x^2*(1+x)/(1-x^2-x^4)) \\ G. C. Greubel, May 01 2017

[fibonacci(n//2) for n in range(61)] # G. C. Greubel, Oct 22 2024

a(n) = a(n-2) + a(n-4).
G.f.: x^2*(1+x)/(1-x^2-x^4). - R. J. Mathar, Sep 27 2008
a(n) = A000045(floor(n/2)). - Johannes W. Meijer, Aug 16 2011

Edited by N. J. A. Sloane, Dec 01 2006

Incorrect formula deleted by Johannes W. Meijer, Aug 16 2011

A031877 Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.

Original entry on oeis.org

8712, 9801, 87912, 98901, 879912, 989901, 8799912, 9899901, 87128712, 87999912, 98019801, 98999901, 871208712, 879999912, 980109801, 989999901, 8712008712, 8791287912, 8799999912, 9801009801, 9890198901, 9899999901, 87120008712, 87912087912, 87999999912

Offset: 1

Eric W. Weisstein

The terms of this sequence are sometimes called palintiples.
All terms are of the form 87...12 = 4*21...78 or 98...01 = 9*10...89. [This was proved by Hoey, 1992. - N. J. A. Sloane, Oct 19 2014] More precisely, they are obtained from concatenated copies of either 8712 or 9801, with 9's inserted "in the middle of" these and/or 0's inserted between the copies these, in a symmetrical way. A008919 lists the reversals, but not in the same order, e.g., R(a(2)) < R(a(1)). - M. F. Hasler, Aug 18 2014
There are 2*Fibonacci(floor((n-2)/2)) terms with n digits (this is A214927 or essentially twice A103609). - Ray Chandler, Oct 11 2017

W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays (1939, page 13); 13th ed. New York: Dover, pp. 14-15, 1987.
G. H. Hardy, A Mathematician's Apology (Cambridge Univ. Press, 1940, reprinted 2000), pp. 104-105 (describes this problem as having "nothing in [it] which appeals much to a mathematician.").

Ray Chandler, Table of n, a(n) for n = 1..10000
Martin Beech, A Computer Conjecture of a Non-Serious Theorem, Mathematical Gazette, 74 (No. 467, March 1990), 50-51.
Patrick De Geest, Palindromic Products of Integers and their Reversals
D. J. Hoey, Palintiples
D. J. Hoey, Palintiples [Cached copy]
Benjamin V. Holt, Some General Results and Open Questions on Palintiple Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 14, Paper A42, 2014.
Benjamin V. Holt, A Determination of Symmetric Palintiples, arXiv:1410.2356 [math.NT], 2014.
Benjamin V. Holt, Families of Asymmetric Palintiples Constructed from Symmetric and Shifted-Symmetric Palintiples, arXiv:1412.0231 [math.NT], 2014.
L. H. Kendrick, Young Graphs: 1089 et al, arXiv:1410.0106 [math.NT], 2014.
L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7.
Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132.
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]
Eric Weisstein's World of Mathematics, Reversal.

See A008919 for reversals (this is the main entry for the problem).
Cf. A169824, A214927, A103609.
Union of A222814 and A222815.
Subsequence of A118959.

a031877_list = [x | x <- [1..], x `mod` 10 > 0,
                    let x' = a004086 x, x' /= x && x `mod` x' == 0]
-- Reinhard Zumkeller, Jul 15 2013

fQ[n_] := Block[{id = IntegerDigits@n}, Mod[n, FromDigits@ Reverse@id] == 0 && n != FromDigits@ Reverse@ id && Mod[n, 10] > 0]; k = 1; lst = {}; While[k < 10^9, If[fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Jun 11 2010 *)
okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[Flatten[ {(4*198)#,(9*99)#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0,1},n], okQ],{n,12}]]]] (* Harvey P. Dale, Jul 03 2013 *)

is_A031877(n)={n%10 && n%A004086(n)==0 && n>A004086(n)} \\ M. F. Hasler, Aug 18 2014

A031877 = []
for n in range(1,10**7):
    if n % 10:
        s1 = str(n)
        s2 = s1[::-1]
        if s1 != s2 and not n % int(s2):
            A031877.append(n) # Chai Wah Wu, Sep 05 2014

a(n) = A004086(a(n))*[9/(a(n)%10)], where [...]=9 if a(n) ends in "1" and [...]=4 if a(n) ends in "2". - M. F. Hasler, Aug 18 2014

More terms from Jud McCranie, Aug 15 2001

More terms from Sam Mathers, Aug 18 2014

A008918 Numbers k such that 4*k = (k written backwards), k > 0.

Original entry on oeis.org

2178, 21978, 219978, 2199978, 21782178, 21999978, 217802178, 219999978, 2178002178, 2197821978, 2199999978, 21780002178, 21978021978, 21999999978, 217800002178, 217821782178, 219780021978, 219978219978, 219999999978, 2178000002178, 2178219782178

Offset: 1

N. J. A. Sloane

There are Fibonacci(floor((k-2)/2)) terms with k digits (this is essentially A103609). - Ray Chandler, Oct 12 2017

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 41-42.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 200 terms from Vincenzo Librandi)
C. A. Van Cott, The Integer Hokey Pokey, Math Horizons, Vol. 28, pp. 24-27, November 2020.
L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7.
N. J. A. Sloane, 2178 And All That, arXiv:1307.0453 [math.NT], 2013; Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Cf. A001232, A193434, A008918, A008919, A222814, A222815, A031877.

Rest@Select[FromDigits /@ Tuples[{0, 198}, 11], IntegerDigits[4*#] == Reverse@IntegerDigits[#] &] (* Arkadiusz Wesolowski, Aug 14 2012 *)
okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; 198#&/@ Flatten[ Table[FromDigits/@Select[Tuples[{0,1},n],okQ],{n,20}]] (* Harvey P. Dale, Jul 03 2013 *)

rev(n) = (eval(concat(Vecrev(Str(n)))));
isok(n) = rev(n) == 4*n; \\ Michel Marcus, Sep 13 2015

Theorem (David W. Wilson): a(n) = 2*A001232(n).

Corrected and extended by David W. Wilson Aug 15 1996, Dec 15 1997

a(20)-a(21) from Arkadiusz Wesolowski, Aug 14 2012

A008919 Numbers k such that k written backwards is a nontrivial multiple of k.

Original entry on oeis.org

1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, 10891089, 10999989, 21782178, 21999978, 108901089, 109999989, 217802178, 219999978, 1089001089, 1098910989, 1099999989, 2178002178, 2197821978, 2199999978, 10890001089

Offset: 1

N. J. A. Sloane

There are 2*Fibonacci(floor((n-2)/2)) terms with n digits (this is A214927 or essentially twice A103609). - N. J. A. Sloane, Mar 20 2013
All terms are made of "symmetric" concatenations of 1089 and/or 2178, with an arbitrary numbers of 9's inserted in the middle of these and 0's inserted between them. See A031877 for the reversals and further information: union of the two, sequences "made of" 1089 or 2178 only. - M. F. Hasler, Jun 23 2019
Also: 99 times A061852: numbers that are palindromic, have only digits in {0, 1} or in {0, 2}, and no isolated ("single") digit. - M. F. Hasler, Oct 17 2022

W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays (1939, page 13); 13th ed. New York: Dover, pp. 14-15, 1987.
Gardiner, Anthony, and A. D. Gardiner. Discovering mathematics: The art of investigation. Oxford University Press, 1987.
G. H. Hardy, A Mathematician's Apology (Cambridge Univ. Press, 1940, reprinted 2000), pp. 104-105 (describes this problem as having "nothing in [it] which appeals much to a mathematician").
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 400 terms from Vincenzo Librandi)
Yannis Almirantis and Wentian Li, Iterative Digital Reversion: a simple algorithm deploying a complex phenomenology related to the '1089 effect', ResearchGate, 2024. See p. 17.
Martin Beech, A Computer Conjecture of a Non-Serious Theorem, Mathematical Gazette, 74 (No. 467, March 1990), 50-51.
Patrick De Geest, Palindromic Products of Integers and their Reversals
D. J. Hoey, Palintiples
D. J. Hoey, Palintiples [Cached copy]
Benjamin V. Holt, Some General Results and Open Questions on Palintiple Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 14, Paper A42, 2014.
Benjamin V. Holt, Derived Palintiple Families and Their Palinomials, arXiv:1410.2356 [math.NT], 2014.
Benjamin V. Holt, Families of Asymmetric Palintiples Constructed from Symmetric and Shifted-Symmetric Palintiples, arXiv:1412.0231 [math.NT], 2014.
Benjamin V. Holt, Finding permultiples of a known base and multiplier, Integers (2025) Vol. 25, Art. No. A13. See p. 23, also arXiv:2411.10859 [math.CO], 2024.
Benjamin V. Holt, On the Reflective Symmetry of the Mother Graph, arXiv:2504.17158 [math.CO], 2025. See p. 28.
Benjamin V. Holt, A Multigraph Characterization of Permutiple Strings, arXiv:2505.11414 [math.CO], 2025. See p. 17.
L. H. Kendrick, Young Graphs: 1089 et al., arXiv:1410.0106 [math.NT], 2014.
L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7.
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.
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]
R. Webster and G. Williams, On the Trail of Reverse Divisors: 1089 and All that Follow, Mathematical Spectrum, Applied Probability Trust, Sheffield, Vol. 45, No. 3, 2012/2013, pp. 96-102.
Eric Weisstein's World of Mathematics, Reversal

Cf. A001232 (9k = R(k)), A004086 (R(n): reverse), A008918 (4k = R(k)), A214927, A103609 (Fibonacci([n/2])). Reversals are in A031877.

a008919 n = a008919_list !! (n-1)
a008919_list = [x | x <- [1..],
                    let (x',m) = divMod (a004086 x) x, m == 0, x' > 1]
-- Reinhard Zumkeller, Feb 03 2012

Reap[ Do[ If[ Reverse[ IntegerDigits[n]] == IntegerDigits[4*n], Print[n]; Sow[n]]; If[ Reverse[ IntegerDigits[n + 11]] == IntegerDigits[9*(n + 11)], Print[n + 11]; Sow[n + 11]], {n, 78, 2*10^10, 100}]][[2, 1]] (* Jean-François Alcover, Jun 19 2012, after David W. Wilson, assuming n congruent to 78 or 89 mod 100 *)
okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[ Flatten[ {99#, 198#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0,1},n], okQ],{n,10}]]]] (* Harvey P. Dale, Jul 03 2013 *)

is_A008919(n,r=A004086(n))={n>r && n%r==0} \\ M. F. Hasler, Jun 23 2019

If reverse(n) = k*n in base 10, then k = 1, 4 or 9 [Klosinski and Smolarski]. Hence A008919 is the union of A001232 and A008918. - David W. Wilson

a(n) = 99*A061852(n). - M. F. Hasler, Oct 17 2022

Corrected and extended by David W. Wilson Aug 15 1996, Dec 15 1997

A222814 Numbers (not ending in 0) which are 9 times their digit-reversal.

Original entry on oeis.org

9801, 98901, 989901, 9899901, 98019801, 98999901, 980109801, 989999901, 9801009801, 9890198901, 9899999901, 98010009801, 98901098901, 98999999901, 980100009801, 980198019801, 989010098901, 989901989901, 989999999901, 9801000009801, 9801989019801, 9890100098901

Offset: 1

N. J. A. Sloane, Mar 11 2013

There are Fibonacci(floor((n-2)/2)) terms with n digits (this is essentially A103609). - Ray Chandler, Oct 12 2017

Ray Chandler, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Equals 9*A001232.

okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[ Flatten[ (9*99)#&/@Flatten[Table[FromDigits/@Select[Tuples[{0,1},n],okQ],{n,12}]]]] (* Harvey P. Dale, Jul 03 2013 *)

A226916 Number of (17,11)-reverse multiples with n digits.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 2, 2, 3, 3, 5, 4, 7, 6, 10, 9, 15, 13, 22, 19, 32, 28, 47, 41, 69, 60, 101, 88, 148, 129, 217, 189, 318, 277, 466, 406, 683, 595, 1001, 872, 1467, 1278, 2150, 1873, 3151, 2745, 4618, 4023, 6768, 5896, 9919, 8641, 14537, 12664, 21305, 18560, 31224, 27201, 45761

Offset: 0

N. J. A. Sloane, Jun 24 2013

Comment from Emeric Deutsch, Aug 21 2016 (Start):
Given an increasing sequence of positive integers S = {a0, a1, a2, ... }, let
          F(x) = x^{a0} + x^{a1} + x^{a2} + ... .
Then the g. f. for the number of palindromic compositions of n with parts in S is (see Hoggatt and Bicknell, Fibonacci Quarterly, 13(4), 1975):
      (1 + F(x))/(1 - F(x^2))
Playing with this, I have found easily that
1. number of palindromic compositions of n into {3,4,5,...} = A226916(n+4);
2. number of palindromic compositions of n into {1,4,7,10,13,...} = A226916(n+6);
3. number of palindromic compositions of n into {1,4} = A226517(n+10);
4. number of palindromic compositions of n into {1,5} = A226516(n+11).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. E. Hogatt, M. Bicknell, Palindromic Compositions, Fib. Quart. 13 (4) (1975) 350-356
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,1).

Cf. A214927, A226516, A226517.

CoefficientList[Series[x^4 (1 - x^2 + x^3 + x^4) / (1 - x^2 - x^6), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 16 2013 *)

G.f.: x^4*(1+x)*(1-x+x^3)/(1-x^2-x^6).

a(2n) = A058278(n-1). a(2n+1)=A000930(n-3). - R. J. Mathar, Dec 13 2022

A101705 Numbers n such that n = 12*reversal(n).

Original entry on oeis.org

0, 540, 5940, 54540, 59940, 540540, 599940, 5400540, 5454540, 5945940, 5999940, 54000540, 54594540, 59405940, 59999940, 540000540, 540540540, 545454540, 545994540, 594005940, 594545940, 599459940, 599999940, 5400000540, 5405940540, 5454054540, 5459994540, 5940005940, 5945945940, 5994059940, 5999999940

Offset: 1

Farideh Firoozbakht, Jan 02 2005

60 divides all terms of the sequence. For all nonnegative integers m and n all numbers of the form f(m,n) = (100*(6*10^m - 1)+ 40)*(10^((m + 2)*n) - 1)/(10^(m + 2) - 1) are in the sequence, in fact f(m,n) = (5.(9)(m))(n).0 where dot between numbers means concatenation and "(r)(t)" means number of r's is t. f(m,1) = 100*(6*10^m - 1)+ 40 = 5.(9)(m).40; f(0,1) = 540, f(1,1) = 5940, f(2,1)=59940, etc. f(m,2) = 5.(9)(m).50(9)(m).40; f(0,2) = 54540, f(1,2) = 5945940, etc. Let g(s,t,r) = s*(10^((L+t)(1+r))-1)/(10^(L+t)-1) where L = number of digits of s. If s is in the sequence then all numbers of the form g(s,t,r) for nonnegative integers t and r are in the sequence (the function g is the same function that has been defined in the sequence A101704). If n and m are nonnegative integers then g(n,0,m) = (n)(m+1) for example g(13,0,3) = (13)(4) = 13131313.

			g(540,0,5)= (540)(6) = 540540540540540540 is in the sequence because reversal(540540540540540540) = 45045045045045045 and 12*45045045045045045 = 540540540540540540.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A101704, A101706, A001232, A008918.

Do[If[n == 12*FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 0, 6000000000, 60}]
Select[Range[0,6*10^9,60],#==12IntegerReverse[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 12 2017 *)

def A101705(n):
    if n == 1: return 0
    a = 1<Chai Wah Wu, Jul 23 2024

a(n) = 540*A057148. - Ray Chandler, Oct 09 2017

a(25)-a(31) from Max Alekseyev, Aug 18 2013

A222815 Numbers (not ending in 0) which are 4 times their digit-reversal.

Original entry on oeis.org

8712, 87912, 879912, 8799912, 87128712, 87999912, 871208712, 879999912, 8712008712, 8791287912, 8799999912, 87120008712, 87912087912, 87999999912, 871200008712, 871287128712, 879120087912, 879912879912, 879999999912, 8712000008712, 8712879128712, 8791200087912

Offset: 1

N. J. A. Sloane, Mar 11 2013

There are Fibonacci(floor((n-2)/2)) terms with n digits (this is essentially A103609). - Ray Chandler, Oct 12 2017

Ray Chandler, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Equals 4*A008918.

okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[Flatten[ (4*198)#&/@Flatten[Table[FromDigits/@Select[Tuples[{0,1},n],okQ],{n,12}]]]] (* Harvey P. Dale, Jul 03 2013 *)

cp's OEIS Frontend

A039684 Erroneous version of A001232.

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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.

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A103609 Fibonacci numbers repeated (cf. A000045).

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A031877 Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.

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A008918 Numbers k such that 4*k = (k written backwards), k > 0.

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A008919 Numbers k such that k written backwards is a nontrivial multiple of k.

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A222814 Numbers (not ending in 0) which are 9 times their digit-reversal.

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