1,3

n is a palindrome (i.e., a(k) = n for some k) if and only if n = A004086(n). - Reinhard Zumkeller, Mar 10 2002

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

The order has been reduced from 49 to 3; see the Cilleruelo-Luca and Cilleruelo-Luca-Baxter links. - Jonathan Sondow, Nov 27 2017

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 it is 9 * 10^(floor((d - 1)/2)). - N. J. A. Sloane, Dec 06 2015

Sequence A033665 tells how many iterations of the Reverse-then-add function A056964 are needed to reach a palindrome; numbers for which this will never happen are Lychrel numbers (A088753) or rather Kin numbers (A023108). - M. F. Hasler, Apr 13 2019

This sequence is an additive basis of order 3, see Cilleruelo, Luca, & Baxter and Sigg. - Charles R Greathouse IV, Apr 08 2025

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 277.

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.

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 50-52.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 120.

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).

T. D. Noe, List of first 19999 palindromes: Table of n, a(n) for n = 1..19999

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

William D. Banks, Derrick N. Hart, and Mayumi Sakata, Almost all palindromes are composite, Math. Res. Lett., Vol. 11, No. 5-6 (2004), pp. 853-868.

William D. Banks, Every natural number is the sum of forty-nine palindromes, arXiv:1508.04721 [math.NT], 2015; Integers, 16 (2016), article A3.

Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.

Patrick De Geest, World of Numbers.

Kritkhajohn Onphaeng, Tammatada Khemaratchatakumthorn, Phakhinkon Napp Phunphayap, and Prapanpong Pongsriiam, Exact Formulas for the Number of Palindromes in Certain Arithmetic Progressions, Journal of Integer Sequences, Vol. 27 (2024), Article 24.4.8. See p. 2.

Phakhinkon Phunphayap and Prapanpong Pongsriiam, Reciprocal sum of palindromes, arXiv:1803.00161 [math.CA], 2018.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Prapanpong Pongsriiam and Kittipong Subwattanachai, Exact Formulas for the Number of Palindromes up to a Given Positive Integer, Intl. J. of Math. Comp. Sci. (2019) 14:1, 27-46.

E. A. Schmidt, Positive Integer Palindromes. [Cached copy at the Wayback Machine]

Markus Sigg, On a conjecture of John Hoffman regarding sums of palindromic numbers, arXiv preprint (2015). arXiv:1510.07507 [math.NT]

Eric Weisstein's World of Mathematics, Palindromic Number.

Wikipedia, Palindromic number.

Index entries for sequences that are an additive basis, order 3.

A136522(a(n)) = 1.

A178788(a(n)) = 0 for n > 10. - Reinhard Zumkeller, Jun 30 2010

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

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

Sum_{n>=2} 1/a(n) = A118031. - Amiram Eldar, Oct 17 2020

a(n) = (floor(d(n)/(c(n)*9 + 1)))*10^A055642(d(n)) + A004086(d(n)) where b(n, k) = ceiling(log((n + 1)/k)/log(10)), c(n) = b(n, 2) - b(n, 11) and d(n) = (n - A086573(b(n*(2 - c(n)), 2) - 1)/2 - 1). - Alan Michael Gómez Calderón, Mar 11 2025

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

# Alternative: 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

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 b > 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 *)

Select[Range[10^3], PalindromeQ] (* Michael De Vlieger, Nov 27 2017 *)

nLP[cn_Integer]:=Module[{s, len, half, left, pal, fdpal}, s=IntegerDigits[cn]; len=Length[s]; half=Ceiling[len/2]; left=Take[s, half]; pal=Join[left, Reverse[ Take[left, Floor[len/2]]]]; fdpal=FromDigits[pal]; Which[cn==9, 11, fdpal>cn, fdpal, True, left=IntegerDigits[ FromDigits[left]+1]; pal=Join[left, Reverse[Take[left, Floor[len/2]]]]; FromDigits[pal]]]; NestList[nLP, 0, 100] (* Harvey P. Dale, Dec 10 2024 *)

(PARI) is_A002113(n)=Vecrev(n=digits(n))==n \\ M. F. Hasler, Nov 17 2008, updated Apr 26 2014, Jun 19 2018

(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

(PARI) inv_A002113(P)={P\(P=10^(logint(P+!P, 10)\/2))+P} \\ index n of palindrome P = a(n), much faster than above: no sum is needed. - M. F. Hasler, Sep 09 2018

(PARI) A002113(n, L=logint(n, 10))=(n-=L=10^max(L-(n<11*10^(L-1)), 0))*L+fromdigits(Vecrev(digits(if(n<L, n, n\10)))) \\ M. F. Hasler, Sep 11 2018

(Python) # edited by M. F. Hasler, Jun 19 2018

def A002113_list(nMax):

mlist=[]

for n in range(nMax+1):

mstr=str(n)

if mstr==mstr[::-1]:

mlist.append(n)

return mlist # Bill McEachen, Dec 17 2010

(Python)

from itertools import chain

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

(Python)

from itertools import chain, count

A002113 = chain(k for k in count(0) if str(k) == str(k)[::-1])

print([next(A002113) for k in range(60)]) # Jan P. Hartkopf, Apr 10 2021

(Python) is_A002113 = lambda n: (s:=str(n))[::-1]==s # M. F. Hasler, May 23 2024

(Python)

from math import log10, floor

def A002113(n):

if n < 2: return 0

P = 10**floor(log10(n//2)); M = 11*P

s = str(n - (P if n < M else M-P))

return int(s + s[-2 if n < M else -1::-1]) # M. F. Hasler, Jun 06 2024

(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

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

(SageMath)

[n for n in (0..515) if Word(n.digits()).is_palindrome()] # Peter Luschny, Sep 13 2018

(GAP) Filtered([0..550], n->ListOfDigits(n)=Reversed(ListOfDigits(n))); # Muniru A Asiru, Mar 08 2019

(Scala) def palQ(n: Int, b: Int = 10): Boolean = n - Integer.parseInt(n.toString.reverse) == 0

(0 to 999).filter(palQ(_)) // Alonso del Arte, Nov 10 2019

Subsequence of A061917 and A221221.

A110745 is a subsequence.

Union of A056524 and A056525.

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 nonprimes: A032350.

Palindromic-pi: A136687.

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

Palindromic floor function: A261423, also A261424. Palindromic ceiling: A262038.

Cf. A004086 (read n backwards), A064834, A118031, 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.

Sequence in context: A297271 A110751 A147882 * A227858 A335779 A240601

Adjacent sequences: A002110 A002111 A002112 * A002114 A002115 A002116

nonn,base,easy,nice,core

approved