A094707 -id:A094707 - cp's OEIS frontend

A001232 Numbers k such that 9*k = (k written backwards), k > 0.

1089, 10989, 109989, 1099989, 10891089, 10999989, 108901089, 109999989, 1089001089, 1098910989, 1099999989, 10890001089, 10989010989, 10999999989, 108900001089, 108910891089, 109890010989, 109989109989, 109999999989, 1089000001089, 1089109891089

Offset: 1

N. J. A. Sloane and Simon Plouffe

This sequence contains the least n-digit non-palindromic number which is a factor of its reversal. Quotient is always 9. - Lekraj Beedassy, Jun 11 2004. (But it contains many other numbers as well. - N. J. A. Sloane, Jul 02 2013)
Nonzero fixed points of the map which sends x to x - reverse(x) if that is nonnegative, otherwise to x + reverse(x). - Sébastien Dumortier, Nov 05 2006. (Clarified comment, see A124074. - Ray Chandler, Oct 11 2017)
Numbers k such that reversal(k)=reversal(k+reversal(k)). Also numbers k such that reversal(k)=reversal(10*k-reversal(k)). - Farideh Firoozbakht, Jun 11 2010
From M. F. Hasler, Oct 04 2022: (Start)
(1) The first digit of any term must be 1, otherwise multiplication by 9 yields one more digit. For the same reason, no "overflow" must occur from the second to the first digit, so the last digit must be 9.
(2) Continuing the reasoning "from right to left" implies that the trailing nonzero digits must be ...9*89, where 9* means any nonnegative number of consecutive digits 9, preceded by a digit 0, which must be preceded by a digit 1. This implies that the initial and also final digits of any term must be 109*89. We might call a term of this form a "primitive" term. So there is exactly one primitive term b(k) = 11*10^(k-2)-11 with k digits, for all k >= 4.
(3) All terms of the sequence are a "symmetric" concatenation of such b(k)'s, "spaced out" with any number of digits 0, also in a symmetrical way: For any n >= 1, let k = (k[1], ..., k[n]) with k[n+1-j] = k[j] >= 4, and m = (m[1], ..., m[n-1]) (possibly of length 0) with m[n-j] = m[j] >= 0, then N = concat(b(k[j])*10^m[j], 1 <= j < n; k[n]) is a term of the sequence, and this yields all terms of the sequence. (For example, with 1089 we also have 1089{0...0}1089 and 1089,001089,001089, etc.) (End)

			1089*9 = 9801.

H. Camous, Jouer Avec Les Maths, "Cardinaux Réversibles", Section I, Problem 6, pp. 27, 37-38; Les Editions d'Organisation, Paris, 1984.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 41.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, under #1089.

Ray Chandler, Table of n, a(n) for n = 1..10000
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]
Simon Weisgerber, Value Judgments in Mathematics: GH Hardy and the (Non-)seriousness of Mathematical Theorems, Global Phil. (2024) Vol. 34, Art. No. 1. See p. 8.

Cf. A008918, A008919, A193434, A222814, A222815, A031877, A124074.

Cf. A002275, A002283, A004086, A094707.

Rest@Select[FromDigits /@ Tuples[{0, 99}, 11], IntegerDigits[9*#] == Reverse@IntegerDigits[#] &] (* Arkadiusz Wesolowski, Aug 14 2012 *)
okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; 99#&/@Flatten[Table[ FromDigits/@ Select[Tuples[{0,1},n],okQ],{n,20}]] (* Harvey P. Dale, Jul 03 2013 *)

isok(n) = 9*n == eval(concat(Vecrev(Str(n)))); \\ Michel Marcus, Feb 21 2015

{A001232_row(n, L(v, s=0)=for(i=1, #v, s*=10^v[i]; i%2 && s+=10^v[i]\900); s)=if(n<4, [], L, Set(apply(L, self()(n, 0)))*99, L=List([[n]]); for(k=4, n\2, listput(L,[k,n-2*k,k]); for(p=0, n\2-k, foreach(self()(n-(k+p)*2, 0), M, listput(L, concat([[k, p], M, [p, k]]))))); L)} \\ List of n-digit terms. - M. F. Hasler, Oct 04 2022
concat(apply(A001232_row, [1..14]))

def A001232_row(n, r=11): # list of n-digit terms
    L = [] if n<4 else [[n]]
    for L1 in range(4, n//2+1):
        L.append([L1, n-2*L1, L1])
        L.extend([L1,L2]+M+[L2,L1] for L2 in range(n//2-1-L1)
                                     for M in A001232_row(n-(L1+L2)*2, 0))
    if not r: return L
    def f(L, s=0):
        for k,L in enumerate(L):
            s *= 10**L
            if not k%2: s += 10**(L-2)-1
        return r*s
    return sorted(map(f, A001232_row(n, 0))) # M. F. Hasler, Oct 04 2022

Theorem: Terms in this sequence have the form 99*m, where the decimal representation of m contains only 1's and 0's, is palindromic and contains no singleton 1's or 0's. Hence contains Fib(floor(k/2)-1) k-digit terms, k >= 4. - David W. Wilson, Dec 15 1997
a(A094707(n)) = 11*(10^n - 1) = 11*A002283(n) = 99*A002275(n), for n>1. - Lekraj Beedassy, Jun 11 2004. (Restored from history and corrected. - Ray Chandler, Oct 11 2017)
a(n) = 99*A061851(n) = A008918(n)/2. - M. F. Hasler, Oct 06 2022

Corrected and extended by David W. Wilson, Aug 15 1996, Dec 15 1997
a(20)-a(21) from Arkadiusz Wesolowski, Aug 14 2012
a(1..10^4) in b-file double-checked with independent code by M. F. Hasler, Oct 04 2022

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

A142245 Expansion of 2x(6 + 5x) / ((1 - x)(1 - x - x^2)).

Original entry on oeis.org

0, 12, 34, 68, 124, 214, 360, 596, 978, 1596, 2596, 4214, 6832, 11068, 17922, 29012, 46956, 75990, 122968, 198980, 321970, 520972, 842964, 1363958, 2206944, 3570924, 5777890, 9348836, 15126748, 24475606, 39602376, 64078004, 103680402, 167758428, 271438852, 439197302

Offset: 0

Paul Curtz, Sep 18 2008

The generic a(n) = 2*a(n-1)-a(n-3) for this family of recurrences (see the link to the OEIS index) leads directly to a common symmetry of the form a(n+1)-2a(n) = 12, 10, 0, -12, -34, -68, -124,... = 12, 10, -a(n).

Todd Silvestri, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A094707, A022095, A000045, A000032.

a[n_Integer/;n>=0]:=23 Fibonacci[n]+11 LucasL[n]-22 (* Todd Silvestri, Dec 16 2014 *)
LinearRecurrence[{2,0,-1},{0,12,34},40] (* Harvey P. Dale, May 12 2015 *)

concat(0, Vec(2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)) + O(x^50))) \\ Colin Barker, Nov 13 2017

G.f.: 2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)).
a(n) = 10*A094707(2*n) + A094707(2*n+1).
a(n) = 2*A022095(n+3) - 22. - R. J. Mathar, Jul 07 2011
a(n) = 23*F(n)+11*L(n)-22 = 23*A000045(n)+11*A000032(n)-22, where F(n) and L(n) are the n-th Fibonacci and Lucas numbers, respectively. - Todd Silvestri, Dec 16 2014
a(n) = (1/5)*(-110 + (55-23*sqrt(5))*((1-sqrt(5))/2)^n + ((1+sqrt(5))/2)^n*(55+23*sqrt(5))). - Colin Barker, Nov 13 2017

cp's OEIS Frontend

A001232 Numbers k such that 9*k = (k written backwards), k > 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A103609 Fibonacci numbers repeated (cf. A000045).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A142245 Expansion of 2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A142245 Expansion of 2x(6 + 5x) / ((1 - x)(1 - x - x^2)).