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A002113
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Palindromes in base 10.
(Formerly M0484 N0178)
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251
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515
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graph;
refs;
listen;
history;
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internal format)
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OFFSET
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1,3
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COMMENTS
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n is a palindrome (i.e. a(k)=n for some k) iff n = A004086(n). - Reinhard Zumkeller, Mar 10 2002
The g.f. -z*(1+10*z**9+10*z**10+81*z**11+9*z)/(8*z**10-9*z**9-9*z-1)/(z-1)**2 conjectured by Simon Plouffe in his 1992 dissertation is wrong. - N. J. A. Sloane, May 12 2008
A178788(a(n)) = 0 for n > 9. [From Reinhard Zumkeller, Jun 30 2010]
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REFERENCES
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Karl G. Grueber: "Palindrome, Perioden und Chaoten: 66 Streifzuege durch die palindromischen Gefilde" (1997, Deutsch-Taschenbuecher; Bd. 99) ISBN 3-8171-1522-9.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, List of first 19999 palindromes: Table of n, a(n) for n = 1..19999
Patrick De Geest, World of Numbers
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
E. A. Schmidt, Positive Integer Palindromes
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes
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FORMULA
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A136522(a(n)) = 1.
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MAPLE
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read transforms; t0:=[]; for n from 0 to 2000 do if digrev(n) = n then t0:=[op(t0), n]; fi; od: t0;
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MATHEMATICA
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palQ[n_Integer, base_Integer] := Module[{idn=IntegerDigits[n, base]}, idn==Reverse[idn]]; then to generate any base-b sequence: Select[Range[0, 1000], palQ[ #, b]&]
lst = {0}; r = 2; Do[Do[AppendTo[lst, n*10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}]; Do[AppendTo[lst, n*10^IntegerLength[n] + FromDigits@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}], {e, r}]; lst (* Arkadiusz Wesolowski, May 04 2012 *)
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PROG
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(PARI) is_A002113(n) = vecextract(n=Vec(Vecsmall(Str(n))), "-1..1")==n [From M. F. Hasler, Nov 17 2008]
(PARI) is(n)=n=digits(n); for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1 \\ Charles R Greathouse IV, Jan 04 2013
(Python)# A002113.py (replace leading dots with blanks)
mlist=[]
ctr=0
n=0
nmax=input("Enter max n ")
while n<=nmax:
...mstr=str(n)
...if mstr==mstr[::-1]:
......mlist.append(mstr)
......ctr+=1
...n+=1
print(mlist)
print("")
print(ctr)
(Haskell)
a002113 n = a002113_list !! (n-1)
a002113_list = filter ((== 1) . a136522) [1..]
-- Reinhard Zumkeller, Oct 09 2011
(Haskell)
a002113_list = m a056524_list a056525_list where
m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
| otherwise = y : m xs' ys
-- Reinhard Zumkeller, Dec 28 2011
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CROSSREFS
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Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.
Cf. A057148, A118594, A118595, A118596, A118597, A118598, A118599, A118600.
Subsequence of A061917.
Cf. A029742 (complement).
Union of A056524 and A056525.
Sequence in context: A043713 A110751 A147882 * A084982 A110785 A193413
Adjacent sequences: A002110 A002111 A002112 * A002114 A002115 A002116
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KEYWORD
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nonn,base,easy,nice,core
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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