A214927 -id:A214927 - cp's OEIS frontend

A103609 Fibonacci numbers repeated (cf. A000045).

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

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

A091079 Numbers n which when converted to base 5, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

16, 96, 416, 496, 576, 2016, 2496, 2976, 10016, 10416, 12096, 12496, 14976, 50016, 52416, 60096, 62496, 74976, 250016, 252016, 260416, 262416, 300096, 302096, 310496, 312496, 360576, 374976, 1250016, 1262016, 1300416, 1312416, 1500096, 1512096, 1550496

Offset: 1

Chuck Seggelin, Dec 18 2003

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

n/m must be either 2 or 4. - Robert Israel, Apr 22 2021

			a(1) = 16 because: 16 in base 5 is 31; 31 reversed is 13; 13 converted back to base 10 is 8 and 16 mod 8 = 0.

Robert Israel, Table of n, a(n) for n = 1..10000
C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link?]

Cf. A091077 (same in base 3), A091078 (base 4), A091080 (base 6), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).

A078477 Number of rational knots with n crossings and unknotting number = 1 (chiral pairs counted only once).

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 7, 15, 15, 30, 31, 63, 63, 126, 127, 255, 255, 510, 511, 1023, 1023, 2046, 2047, 4095, 4095, 8190, 8191, 16383, 16383, 32766, 32767, 65535, 65535, 131070, 131071, 262143, 262143, 524286, 524287, 1048575, 1048575, 2097150, 2097151

Offset: 3

Ralf Stephan, Jan 03 2003

From Alexander Adamchuk, Nov 16 2009: (Start)
For n>1 a(2n+1) = 2^(n-1) - 1 = A000225(n-1).
For n>1 a(4n) = a(4n+1) - 1 = 2^(2n-1) - 2.
For n>0 a(4n+2) = a(4n+3) = 2^(2n) - 1. (End)

Andrey Zabolotskiy, Table of n, a(n) for n = 3..1000
A. Stoimenow, Generating Functions, Fibonacci Numbers and Rational Knots, Journal of Algebra, 310 (2007), 491-525; arXiv:math/0210174 [math.GT], 2002.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-2).

Cf. A000225, A018240, A089891, A089892, A089797, A051449, A214927, A051450, A078478.

CoefficientList[Series[(2 x^7 + 2 x^6 - x^5 + x^3 - x^2 + x + 1) / ((x-1) (x+1) (x^2+1) (2 x^2-1)), {x, 0, 50}], x] (* Vincenzo Librandi, May 17 2013 *)

Vec(x^3*(1+x-x^2+x^3-x^5+2*x^6+2*x^7)/((1-x)*(1+x)*(1+x^2)*(1-2*x^2)) + O(x^60)) \\ Colin Barker, Dec 26 2015

G.f.: x^3*(1+x-x^2+x^3-x^5+2*x^6+2*x^7) / ((1-x)*(1+x)*(1+x^2)*(1-2*x^2)).

a(n) = 2*a(n-2)+a(n-4)-2*a(n-6) for n>10. - Colin Barker, Dec 26 2015

A222810 Number of n-digit numbers N with distinct digits such that the reversal of N divides N.

Original entry on oeis.org

9, 9, 3, 5, 3, 2, 0, 0, 0

Offset: 1

N. J. A. Sloane, Mar 10 2013

Suggested by A214927.

a(n)=0 for all n > 6.

			Solutions with 1 through 6 digits:
[1, 2, 3, 4, 5, 6, 7, 8, 9],
[10, 20, 30, 40, 50, 60, 70, 80, 90],
[510, 540, 810],
[5610, 5940, 8712, 8910, 9801],
[65340, 87120, 87912],
[659340, 879120],

Cf. A214927, A222809, A222811, A222812.

For the actual numbers see A223080.

import collections
col = []
count = 0
for n in range(0, 9):
    a = 10**n
    stop = 10**(n+1)
    while a < stop:
        b = str(a)
        c = list(b)
        d = c[::-1]
        e = int("".join(c))
        f = int("".join(d))
        counter = collections.Counter(c)
        if e % f == 0 and counter.most_common(1)[0][1] == 1:
            count += 1
            col.append(a)
        a += 1
    print(n+1, " digits: ", count, " elements: ", col)
    count = 0
    col = []
# David Consiglio, Jr., Dec 04 2014

a(8)-a(9) from David Consiglio, Jr., Dec 04 2014

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

A222817 Irregular triangle read by rows: row n gives list of nontrivial reverse multipliers for base n.

Original entry on oeis.org

2, 3, 2, 4, 2, 5, 3, 6, 2, 3, 5, 7, 2, 4, 8, 4, 9, 2, 3, 5, 7, 10, 2, 3, 5, 11, 5, 6, 12, 2, 3, 4, 6, 9, 13, 2, 3, 4, 7, 11, 14, 3, 7, 15, 2, 4, 5, 8, 10, 11, 16, 2, 5, 7, 8, 17, 3, 4, 6, 7, 9, 14, 18, 2, 3, 4, 6, 9, 13, 19

Offset: 3

N. J. A. Sloane, Mar 13 2013

If there is a number m such that the reversal of m in base n is c times m, then c is called a reverse multiplier for n. For example, 2 is a reverse multiplier for base n=5, since 8 (base 10) = 13 (base 5), and 2*8 = 16 (base 10) = 31 (base 5).
The trivial reverse multiplier 1 is excluded.
The last entry in each row is n-1; the number of terms in row n is A222819(n).

			Triangle begins:
  2,
  3,
  2,4,
  2,5,
  3,6,
  2,3,5,7,
  2,4,8,
  4,9,
  2,3,5,7,10,
  2,3,5,11,
  5,6,12,
  2,3,4,6,9,13,
  2,3,4,7,11,14,
  3,7,15,
  ...

N. J. A. Sloane, Table giving n, list of nontrivial reverse multipliers, for n = 3..100
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]
Anne Ludington Young, k-Reverse multiples, Fib. Q., 30 (1992), 126-132.

Cf. A214927, A222818, A222819, A222820.

See A214927 for other cross-references.

A061851 Digital representation of n contains only 1's and 0's, is palindromic and contains no singleton 1's or 0's.

Original entry on oeis.org

11, 111, 1111, 11111, 110011, 111111, 1100011, 1111111, 11000011, 11100111, 11111111, 110000011, 111000111, 111111111, 1100000011, 1100110011, 1110000111, 1111001111, 1111111111, 11000000011, 11001110011, 11100000111, 11110001111, 11111111111, 110000000011, 110001100011

Offset: 1

Henry Bottomley, May 10 2001

The terms can be constructed by gluing together terms from A355280 with their reversal as follows: The terms with odd length L = 2k-1 are given from the k-digit terms of A355280 by replacing the last digit with the reversal of the term. (Equivalently, concatenate with the reversal and delete one of the middle digits.) Terms with an even number L = 2k of digits are given as concatenation(m, reverse(m)) = m*10^L(m) + A004086(m) where m runs over the k-digit terms from A355280, and the (k-1)-digit terms with the 1's complement of the last digit appended. This explains the formula given in CROSSREFS for the number of terms of given length. - M. F. Hasler, Oct 17 2022

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 608 terms from N. J. A. Sloane)
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]

Number of terms with k digits is Fibonacci(floor(k/2)) = A000045(A004526(k)).
Union of this sequence and twice this sequence is A061852 and 99 times that is A008919.
Cf. A214927, A001232, A222814, A008918, A008919, A222813 (terms converted to decimal).
Cf. A355280 (palindromic binary numbers with no digit run of length < 2).

concat(apply( {A061851_row(n)=[fromdigits(binary(m))|m<-A222813_row(n)]}, [1..11])) \\ M. F. Hasler, Oct 17 2022

def A061851_row(n): return [] if n < 2 else [10**n//9] if n < 6 else [
    m*10**(n//2) + A004086(m//10) for m in A355280_row(n//2+1)] if n&1 else [
    m*10**(n//2) + A004086(m) for m in sorted(A355280_row(n//2)+
                        [x*10+1-x%10 for x in A355280_row(n//2-1)])]
# M. F. Hasler, Oct 17 2022

a(n) = A001232(n)/99 = A008918(n)/198.

a(n) = A007088(A222813(n)), where A007088 = write in binary. - M. F. Hasler, Oct 06 2022

A222811 Number of n-digit numbers N such that the reversal of N divides N.

Original entry on oeis.org

9, 18, 111, 212, 1128, 2067, 11123, 20270, 110440, 200971, 1101475, 2003592, 11005388

Offset: 1

N. J. A. Sloane, Mar 10 2013

Suggested by A214927.

Even terms are roughly double the previous term. - David Consiglio, Jr., Mar 22 2013

			Some of the smallest solutions are:
[1, 2, 3, 4, 5, 6, 7, 8, 9] (so a(1) = 9),
[10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99] (so a(2) = 18),
[100, 101, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 200, 202, 212, 220, 222, 232, 242, 252, 262, 272, 282, 292, 300, 303, 313, 323, 330, 333, 343, 353, ...]

David Consiglio, Jr., C++ program
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Cf. A214927, A222809, A222810, A222812.

a(8) - a(11) from David Consiglio, Jr., Mar 22 2013

a(12)-a(13) from Giovanni Resta, Apr 01 2013

cp's OEIS Frontend

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

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A091079 Numbers n which when converted to base 5, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

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A078477 Number of rational knots with n crossings and unknotting number = 1 (chiral pairs counted only once).

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A222810 Number of n-digit numbers N with distinct digits such that the reversal of N divides N.

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