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A002113 Palindromes in base 10.
(Formerly M0484 N0178)
298
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

n is a palindrome (i.e. a(k) = n for some k) if and only if 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. - Reinhard Zumkeller, Jun 30 2010

A064834(a(n)) = 0. - Reinhard Zumkeller, Sep 18 2013

REFERENCES

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).

LINKS

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 séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

E. A. Schmidt, Positive Integer Palindromes

Eric Weisstein's World of Mathematics, Palindromic Number

Index entries for sequences related to palindromes

FORMULA

A136522(a(n)) = 1.

MAPLE

read transforms; t0:=[]; for n from 0 to 2000 do if digrev(n) = n then t0:=[op(t0), n]; fi; od: t0;

MATHEMATICA

palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; (* then to generate any base-b sequence for 1 < b < 37, replace the 10 in the following instruction with b: *) Select[Range[0, 1000], palQ[#, 10] &]

base10Pals = {0}; r = 2; Do[Do[AppendTo[base10Pals, n * 10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}]; Do[AppendTo[base10Pals, n * 10^IntegerLength[n] + FromDigits@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}], {e, r}]; base10Pals (* Arkadiusz Wesolowski, May 04 2012 *)

PROG

(PARI) is_A002113(n)={vecextract(n=digits(n+!n), "-1..1")==n} \\ Use digits(n)=Vec(Str(n)) in older PARI versions. - M. F. Hasler, Nov 17 2008, updated Apr 26 2014

(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

(PARI) a(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])} \\ David A. Corneth, Jun 06 2014

(PARI) \\recursive--feed an element a(n) and it gives a(n+1)

nxt(n)={my(d=digits(n)); i=(#d+1)\2; while(i&&d[i]==9, d[i]=0; d[#d+1-i]=0; i--); if(i, d[i]++; d[#d+1-i]=d[i], d=vector(#d+1); d[1]=d[#d]=1); sum(i=1, #d, 10^(#d-i)*d[i])} \\ David A. Corneth, Jun 06 2014

(PARI)\\feed a(n), returns n.

inv(n)={my(d=digits(n)); q=ceil(#d/2); sum(i=1, q, 10^(q-i)*d[i])+10^floor(#d/2)} \\ David A. Corneth, Jun 18 2014

(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

CROSSREFS

Palindromes in bases 2 through 11: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, this, A029956.

Cf. A057148, A118594, A118595, A118596, A118597, A118598, A118599, A118600.

Subsequence of A061917.

Cf. A029742 (complement).

Union of A056524 and A056525.

Subsequence of A221221.

Sequence in context: A043713 A110751 A147882 * A227858 A240601 A084982

Adjacent sequences:  A002110 A002111 A002112 * A002114 A002115 A002116

KEYWORD

nonn,base,easy,nice,core

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 23 23:07 EDT 2014. Contains 244873 sequences.