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A002113 Palindromes in base 10.
(Formerly M0484 N0178)
407
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

A178788(a(n)) = 0 for n > 9. - Reinhard Zumkeller, Jun 30 2010

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

It seems that if n*reversal(n) is in the sequence then n = 3 or all digits of n are less than 3. - Farideh Firoozbakht, Nov 02 2014

The position of a palindrome within the sequence can be determined almost without calculation: If the palindrome has an even number of digits, prepend a 1 to the front half of the palindrome's digits. If the number of digits is odd, prepend the value of front digit + 1 to the digits from position 2 ... central digit. Examples: 98766789=a(19876), 515=a(61), 8206028=a(9206), 9230329=a(10230). - Hugo Pfoertner, Aug 14 2015

This sequence is an additive basis of order at most 49, see Banks link. - Charles R Greathouse IV, Aug 23 2015

See A262038 for the "next palindrome" and A261423 for the "preceding palindrome" functions. - M. F. Hasler, Sep 09 2015

The number of palindromes with d digits is 10 if d=1, and otherwise is 9*10^(floor((d-1)/2)). - N. J. A. Sloane, Dec 06 2015

REFERENCES

Karl G. Kröber, "Palindrome, Perioden und Chaoten: 66 Streifzüge durch die palindromischen Gefilde" (1997, Deutsch-Taschenbücher; 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

Hunki Baek, Sejeong Bang, Dongseok Kim, Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014.

William D. Banks, Every natural number is the sum of forty-nine palindromes, arXiv:1508.04721 [math.NT], 2015.

Javier Cilleruelo and Florian Luca, Every positive integer is a sum of three palindromes, arXiv: 1602.06208 [math.NT], 2016.

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

Wikipedia, Palindromic_number

Index entries for sequences related to palindromes

FORMULA

A136522(a(n)) = 1.

a(n+1)=A262038(a(n)+1). - M. F. Hasler, Sep 09 2015

MAPLE

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

# Alternatively, to get all palindromes with <= N digits in the list "Res":

N:=5;

Res:= $0..9:

for d from 2 to N do

  if d::even then

    m:= d/2;

    Res:= Res, seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1);

  else

    m:= (d-1)/2;

    Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1);

  fi

od: Res:=[Res]: # Robert Israel, Aug 10 2014.

# A variant: Gets all base-10 palindromes with exactly d digits, in the list "Res"

d:=4:

if d=1 then Res:= [$0..9]:

elif d::even then

    m:= d/2:

    Res:= [seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1)]:

else

    m:= (d-1)/2:

    Res:= [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:

fi:

Res; # N. J. A. Sloane, Oct 18 2015

isA002113 := proc(n)

    simplify(digrev(n) = n) ;

end proc: # R. J. Mathar, Sep 09 2015

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 *)

nthPalindromeBase[n_, b_] := Block[{q = n + 1 - b^Floor[ Log[ b, n + 1 - b^Floor[ Log[ b, n/b]] ]], c = Sum[ Floor[ Floor[ n/((b+1)*b^(k-1) - 1)]/(Floor[ n/((b+1)*b^(k-1) - 1)] - 1/ b)] - Floor[ Floor[ n/(2* b^k - 1)]/(Floor[ n/(2* b^k - 1)] - 1/ b)], {k, Floor[ Log[b, n]] }]}, Mod[q, b]*(b+1)^c*b^Floor[Log[b, q]] + Sum[ Floor[ Mod[q,  b^(k + 1)]/ b^k]* b^(Floor[ Log[b, q]] - k) ( b^(2 k + c) + 1), {k, Floor[ Log[b, q]] }]] (* after the work of Eric A. Schmidt, works for all integer bases > 2 *)

Array[ nthPalindromeBase[ #, 10] &, 61, 0] (* please note that Schmidt uses a different, a more natural and intuitive offset, that of a(1) = 1 *) (* Robert G. Wilson v, Sep 22 2014 and modified Nov 28 2014 *)

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)

import Data.List.Ordered (union)

a002113_list = union a056524_list a056525_list

-- Reinhard Zumkeller, Jul 29 2015, Dec 28 2011

(Python)

from itertools import chain

A002113 = sorted(chain(map(lambda x:int(str(x)+str(x)[::-1]), xrange(1, 10**3)), map(lambda x:int(str(x)+str(x)[-2::-1]), xrange(10**3)))) # Chai Wah Wu, Aug 09 2014

(MAGMA) [n: n in [0..600] | Intseq(n, 10) eq Reverse(Intseq(n, 10))]; // Vincenzo Librandi, Nov 03 2014

CROSSREFS

Palindromes in bases 2 through 11: A006995 and A057148, A014190 and A118594, A014192 and A118595, A029952 and A118596, A029953 and A118597, A029954 and A118598, A029803 and A118599, A029955 and A118600, this sequence, A029956. Also A262065 (base 60), A262069 (subsequence).

Palindromic primes: A002385.

Palindromic-pi: A136687.

Cf. A029742 (complement), A086862 (first differences).

Palindromic floor function: A261423, also A261424.

Union of A056524 and A056525.

Cf. A004086 (read n backwards), A064834, A136522 (characteristic function), A178788.

Ways to write n as a sum of three palindromes: A261132, A261422.

Minimal number of palindromes that add to n using greedy algorithm: A088601.

Minimal number of palindromes that add to n: A261675.

Subsequence of A61917 and A221221.

Subsequence: A110745.

Sequence in context: A266140 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 June 28 03:13 EDT 2016. Contains 274263 sequences.