OFFSET
1,1
COMMENTS
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)
REFERENCES
H. Camous, Jouer Avec Les Maths, "Cardinaux Réversibles", Section I, Problem 6, pp. 27, 37-38; Les Editions d'Organisation, Paris, 1984.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, under #1089.
LINKS
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.
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.
FORMULA
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)
EXAMPLE
1089*9 = 9801.
MATHEMATICA
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 *)
PROG
(PARI) isok(n) = 9*n == eval(concat(Vecrev(Str(n)))); \\ Michel Marcus, Feb 21 2015
(PARI) {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]))
(Python)
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
CROSSREFS
KEYWORD
base,nonn,nice,changed
AUTHOR
EXTENSIONS
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
STATUS
approved