A002113 -id:A002113 - cp's OEIS frontend

A259388 Palindromic numbers in bases 5 and 9 written in base 10.

0, 1, 2, 3, 4, 6, 109, 246, 282, 564, 701, 22386, 32152, 41667, 47653, 48553, 1142597, 1313858, 1412768, 1677684, 12607012902, 19671459008, 20134447808, 24208576998, 24863844904, 26358878059

Offset: 1

Robert G. Wilson v, Jul 16 2015

			246 is in the sequence because 246_10 = 303_9 = 1441_5.

Giovanni Resta, Table of n, a(n) for n = 1..40
Index entries for sequences related to palindromes

Cf. A007632, A007633, A029731, A029804, A029961, A029962, A029963, A029964, A029965, A029966, A029967, A029968, A029969, A029970, A048268, A060792, A097855, A097856, A097928, A097929, A097930, A097931, A099145, A099146, A099165, A182232, A182233, A182234, A250408, A250409, A250410, A250411, A250412, A259374, A259375, A259376, A259377, A259378, A249156, A259380, A259381, A259382, A259383, A259384, A259385, A259386, A259387, A259388, A259389, A259390.

(* first load nthPalindromeBase from A002113 *) palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; k = 0; lst = {}; While[k < 21000000, pp = nthPalindromeBase[k, 9]; If[palQ[pp, 5], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=5; b2=9; lst={};Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 10000000}]; lst (* Vincenzo Librandi, Jul 17 2015 *)

Intersection of A029952 and A029955.

A259389 Palindromic numbers in bases 6 and 9 written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 80, 154, 191, 209, 910, 3740, 5740, 8281, 16562, 16814, 2295481, 2300665, 2350165, 2439445, 2488945, 2494129, 2515513, 7971580, 48307924, 61281793, 69432517, 123427622, 124091822, 124443290, 55854298990, 184314116750, 185794441250, 187195815770, 327925630018, 7264479038060, 27832011695551

Offset: 1

Eric A. Schmidt and Robert G. Wilson v, Jul 17 2015

			209 is in the sequence because 209_10 = 252_9 = 545_6.

Giovanni Resta, Table of n, a(n) for n = 1..57
Index entries for sequences related to palindromes

Cf. A048268, A060792, A097856, A097928, A182232, A259374, A097929, A182233, A259375, A259376, A097930, A182234, A259377, A259378, A249156, A097931, A259380, A259381, A259382, A259383, A259384, A099145, A259385, A259386, A259387, A259388, A259389, A259390, A099146, A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A250408, A250409, A250410, A250411, A099165, A250412.

(* first load nthPalindromeBase from A002113 *) palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; k = 0; lst = {}; While[k < 21000000, pp = nthPalindromeBase[k, 9]; If[palQ[pp, 6], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=6; b2=9; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 1000000}]; lst (* Vincenzo Librandi, Jul 17 2015 *)

Intersection of A029953 and A029955.

A332189 a(n) = 8*(10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

9, 898, 88988, 8889888, 888898888, 88888988888, 8888889888888, 888888898888888, 88888888988888888, 8888888889888888888, 888888888898888888888, 88888888888988888888888, 8888888888889888888888888, 888888888888898888888888888, 88888888888888988888888888888, 8888888888888889888888888888888

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332119 .. A332189 (variants with different "wing" digit 1, ..., 8).
Cf. A332180 .. A332187 (variants with different middle digit 0, ..., 7).

A332189 := n -> 8*(10^(2*n+1)-1)/9+10^n;

Array[8 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]

apply( {A332189(n)=10^(n*2+1)\9*8+10^n}, [0..15])

def A332189(n): return 10**(n*2+1)//9*8+10**n

a(n) = 8*A138148(n) + 9*10^n = A002282(2n+1) + 10^n.
G.f.: (9 - 101*x - 700*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A065206 Numbers which need one 'Reverse and Add' step to reach a palindrome.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 40, 41, 42, 43, 45, 47, 50, 51, 52, 53, 54, 56, 60, 61, 62, 63, 65, 70, 71, 72, 74, 80, 81, 83, 90, 92, 100, 102, 103, 104, 105, 106, 107, 108, 110, 112, 113, 114, 115, 116, 117

Offset: 1

Klaus Brockhaus, Oct 21 2001

The number of steps starts at 0, so palindromes (A002113) are excluded.

Numbers k such that A033665(k) = 1. - Andrew Howroyd, Dec 05 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Reverse and Add!

Cf. A002113, A015976, A033665, A136522, A056964, A029742.

Sequences for 2..12 steps needed are: A065207, A065208, A065209, A065210, A065211, A065212, A065213, A065214, A065215, A065216, A065217.

function revadd_steps(k,stop: integer); var c,n,m,steps,rev: integer; begin n := 0; c := 0; while c < stop do m := n; rev := int_reverse(m); steps := 0; while steps < k and m <> rev do m := m + rev; rev := int_reverse(m); inc(steps); end; if steps = k and m = rev then write(n," "); inc(c); end; inc(n); end; end; revadd_steps(1,66).

a065206 n = a065206_list !! (n-1)
a065206_list = filter ((== 1) . a136522 . a056964) a029742_list
-- Reinhard Zumkeller, Oct 14 2011

Select[Range[10,120],!PalindromeQ[#]&&PalindromeQ[#+IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 14 2017 *)

isok(n,s=1)={for(k=0, s, my(r=fromdigits(Vecrev(digits(n)))); if(r==n, return(k==s)); n += r); 0} \\ Andrew Howroyd, Dec 05 2024

Offset corrected by Harry J. Smith, Oct 13 2009

A319453 Number T(n,k) of partitions of n into exactly k nonzero decimal palindromes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1, 0, 0, 5, 8, 9, 7, 5, 3, 2, 1, 1, 0, 1, 4, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 0, 5, 11, 15, 13, 11, 7, 5, 3, 2, 1, 1

Offset: 0

Alois P. Heinz, Sep 19 2018

Differs from A008284 and from A072233 first at T(10,1) = 0.

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1,  1;
  0, 1, 2,  1,  1;
  0, 1, 2,  2,  1,  1;
  0, 1, 3,  3,  2,  1,  1;
  0, 1, 3,  4,  3,  2,  1, 1;
  0, 1, 4,  5,  5,  3,  2, 1, 1;
  0, 1, 4,  7,  6,  5,  3, 2, 1, 1;
  0, 0, 5,  8,  9,  7,  5, 3, 2, 1, 1;
  0, 1, 4, 10, 11, 10,  7, 5, 3, 2, 1, 1;
  0, 0, 5, 11, 15, 13, 11, 7, 5, 3, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A136522 (for n>0), A319468, A261131, A319469, A319470, A319471, A319472, A319473, A319474, A319475.
Row sums give A091580.
T(2n,n) gives A319454.
Cf. A002113, A008284, A072233, A261132.

p:= proc(n) option remember; local i, s; s:= ""||n;
      for i to iquo(length(s), 2) do if
        s[i]<>s[-i] then return false fi od; true
    end:
h:= proc(n) option remember; `if`(n<1, 0,
     `if`(p(n), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..14);

T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A002113(j)).

Sum_{k=0..3} T(n,k) = A261132(n).

A029951 Even palindromes.

Original entry on oeis.org

0, 2, 4, 6, 8, 22, 44, 66, 88, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 808, 818, 828, 838, 848, 858, 868, 878, 888, 898, 2002, 2112, 2222

Offset: 1

Patrick De Geest

John Cerkan, Table of n, a(n) for n = 1..10000
Patrick De Geest, World!Of Numbers

Cf. A002113 (superset), A062287 (subset), A367807 (halved).

Cf. A045638, A045639, A043040, A045641, A045642, A045643, A045644, A083852.

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Select[Range[2, 10^4, 2], palindromicQ[#] &] (* Alonso del Arte, Feb 23 2012 *)
Select[Range[0, 2222, 2], PalindromeQ] (* Michael De Vlieger, May 12 2017, Version 10.3 *)

A033934 a(n) = (10^n + 1)^2.

Original entry on oeis.org

4, 121, 10201, 1002001, 100020001, 10000200001, 1000002000001, 100000020000001, 10000000200000001, 1000000002000000001, 100000000020000000001, 10000000000200000000001, 1000000000002000000000001, 100000000000020000000000001, 10000000000000200000000000001

Offset: 0

Jeff Burch

The members of this sequence are both perfect squares and palindromes. Therefore A002779 is an infinite sequence. - Ant King, Jun 26 2011

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002779 (palindromic squares), A000290 (squares), A002113 (palindromes).

Cf. A011557, A062397, A066138.

(10^Range[0,20]+1)^2 (* or *) LinearRecurrence[{111,-1110,1000},{4,121,10201},20] (* Harvey P. Dale, Feb 16 2016 *)

my(x='x+O('x^15)); Vec((1210*x^2-323*x+4)/(-1000*x^3+1110*x^2-111*x+1)) \\ Elmo R. Oliveira, Jul 04 2025

a(n) = A062397(n)^2 = A066138(n) + A011557(n).
From Elmo R. Oliveira, Jul 04 2025: (Start)
G.f.: (4 - 323*x + 1210*x^2)/((1-x)*(1-10*x)*(1-100*x)).
E.g.f.: exp(x)*(1 + 2*exp(9*x) + exp(99*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3). (End)

Better description from Henry Bottomley, Dec 07 2001

More terms from Harvey P. Dale, Feb 16 2016

A260255 Numbers that can be written as the sum of two nonnegative palindromes in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Reinhard Zumkeller, Jul 21 2015

More than the usual number of terms are shown in order to distinguish this from A261906. - N. J. A. Sloane, Sep 09 2015

A260254(a(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Chai Wah Wu, Fraction of first n numbers that are terms of this sequence, for n < 10^8, Sep 15 2015

Cf. A035137 (complement), A260254, A002113.

111 is a member of this sequence but not of A261906. A213879 lists the differences.

a260255 n = a260255_list !! (n-1)
a260255_list = filter ((> 0) . a260254) [0..]

palQ[n_Integer, base_Integer] := Block[{}, Reverse[idn = IntegerDigits[n, base]] == idn]; Take[ Union[ Plus @@@ Tuples[ Select[ Range[0, 100], palQ[#, 10] &], 2]], 90] (* Robert G. Wilson v, Jul 22 2015 *)

A332119 a(n) = (10^(2n+1)-1)/9 + 8*10^n.

Original entry on oeis.org

9, 191, 11911, 1119111, 111191111, 11111911111, 1111119111111, 111111191111111, 11111111911111111, 1111111119111111111, 111111111191111111111, 11111111111911111111111, 1111111111119111111111111, 111111111111191111111111111, 11111111111111911111111111111, 1111111111111119111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107649 = {1, 4, 26, 187, 226, 874, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (1)9(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11911...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077795-1)/2 = A107649: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
Cf. A332112 .. A332118 (variants with different middle digit 2, ..., 8).

A332119 := n -> (10^(2*n+1)-1)/9+8*10^n;

Array[(10^(2 # + 1)-1)/9 + 8*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,1],{9},PadRight[{},n,1]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{9,191,11911},20] (* Harvey P. Dale, Mar 30 2024 *)

apply( {A332119(n)=10^(n*2+1)\9+8*10^n}, [0..15])

def A332119(n): return 10**(n*2+1)//9+8*10**n

a(n) = A138148(n) + 9*10^n = A002275(2n+1) + 8*10^n.
G.f.: (9 - 808*x + 700*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A351712 Numbers whose minimal (or greedy) Lucas representation (A130310) is palindromic.

Original entry on oeis.org

0, 2, 6, 9, 13, 20, 24, 31, 49, 56, 64, 78, 100, 125, 136, 150, 158, 169, 201, 237, 252, 324, 342, 364, 378, 396, 404, 422, 444, 523, 581, 606, 650, 708, 845, 874, 910, 932, 961, 975, 1004, 1040, 1048, 1077, 1113, 1135, 1164, 1366, 1460, 1500, 1572, 1666, 1692, 1786

Offset: 1

Amiram Eldar, Feb 17 2022

A000211(n) = Lucas(n) + 2 is a term for all n > 2, since the representation of Lucas(n) + 2 is 10...01 with n-1 0's between the two 1's.

			The first 10 terms are:
   n  a(n) A130310(a(n))
   ---------------------
   1   0               0
   2   2               1
   3   6            1001
   4   9           10001
   5  13          100001
   6  20         1000001
   7  24         1001001
   8  31        10000001
   9  49       100000001
  10  56       100010001

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes.

Cf. A000032, A000211, A130310.
Subsequence of A054770.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351717.

lucasPalQ[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; PalindromeQ[IntegerDigits[Total[2^s], 2]]]; Select[Range[0, 2000], lucasPalQ]

cp's OEIS Frontend

A259388 Palindromic numbers in bases 5 and 9 written in base 10.

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A259389 Palindromic numbers in bases 6 and 9 written in base 10.

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A332189 a(n) = 8*(10^(2n+1)-1)/9 + 10^n.

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A065206 Numbers which need one 'Reverse and Add' step to reach a palindrome.

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ARIBAS

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A319453 Number T(n,k) of partitions of n into exactly k nonzero decimal palindromes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A029951 Even palindromes.

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Mathematica

A033934 a(n) = (10^n + 1)^2.

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A260255 Numbers that can be written as the sum of two nonnegative palindromes in base 10.

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