A002113 -id:A002113 - cp's OEIS frontend

A259390 Palindromic numbers in bases 7 and 9 written in base 10.

0, 1, 2, 3, 4, 5, 6, 8, 40, 50, 100, 164, 200, 264, 300, 328, 400, 2000, 3550, 8200, 10252, 14510, 14762, 22800, 45600, 164900, 201720, 400200, 532900, 555013, 738100, 2756120, 2913368, 3344352, 3501600, 4084000, 12990350, 22674550, 194062432, 1684866370, 2225211080, 13575144288, 15127811455, 20404027400, 20537111057, 22668403353, 30862471355, 83714515310, 84668107250, 796259955485, 1202029647736, 2088800185930, 20268849562000

Offset: 1

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

			264 is in the sequence because 264_10 = 323_9 = 525_7.

Giovanni Resta, Table of n, a(n) for n = 1..86
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, 7], AppendTo[lst, pp]; Print[pp]]; k++]; lst

Intersection of A029954 and A029955.

A332120 a(n) = 2(10^(2n+1)-1)/9 - 210^n.

Original entry on oeis.org

0, 202, 22022, 2220222, 222202222, 22222022222, 2222220222222, 222222202222222, 22222222022222222, 2222222220222222222, 222222222202222222222, 22222222222022222222222, 2222222222220222222222222, 222222222222202222222222222, 22222222222222022222222222222, 2222222222222220222222222222222

Offset: 0

M. F. Hasler, Feb 09 2020

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

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332130 .. A332190 (variants with different repeated digit 3, ..., 9).
Cf. A332121 .. A332129 (variants with different middle digit 1, ..., 9).

A332120 := n -> 2*((10^(2*n+1)-1)/9-10^n);

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

apply( {A332120(n)=(10^(n*2+1)\9-10^n)*2}, [0..15])

def A332120(n): return (10**(n*2+1)//9-10**n)*2

a(n) = 2*A138148(n) = A002276(2n+1) - 2*10^n.
G.f.: 2*x*(101 - 200*x)/((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.
E.g.f.: 2*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A332190 a(n) = 10^(2n+1) - 1 - 9*10^n.

Original entry on oeis.org

0, 909, 99099, 9990999, 999909999, 99999099999, 9999990999999, 999999909999999, 99999999099999999, 9999999990999999999, 999999999909999999999, 99999999999099999999999, 9999999999990999999999999, 999999999999909999999999999, 99999999999999099999999999999, 9999999999999990999999999999999

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), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332180 (variants with different repeated digit 2, ..., 8).
Cf. A332191 .. A332197, A181965 (variants with different middle digit 1, ..., 8).

Maple
```
A332190 := n -> 10^(2*n+1)-1-9*10^n;
```

Array[10^(2 # + 1)-1-9*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{0,909,99099},20] (* Harvey P. Dale, May 28 2021 *)

apply( {A332190(n)=10^(n*2+1)-1-9*10^n}, [0..15])

def A332190(n): return 10**(n*2+1)-1-9*10^n

a(n) = 9*A138148(n) = A002283(2n+1) - A011557(n).
G.f.: 9*x*(101 - 200*x)/((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.

A050250 Number of nonzero palindromes less than 10^n.

Original entry on oeis.org

9, 18, 108, 198, 1098, 1998, 10998, 19998, 109998, 199998, 1099998, 1999998, 10999998, 19999998, 109999998, 199999998, 1099999998, 1999999998, 10999999998, 19999999998, 109999999998, 199999999998, 1099999999998, 1999999999998, 10999999999998

Offset: 1

Eric W. Weisstein, Dec 11 1999

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dr. Math, Palindromic Numbers.
Dr. Math, Palindromic Numbers.
G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Palindromic Number.
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A002113, A002778, A050683.

A050250List := proc(len);  local s, egf, ser; s:= 11/(2*sqrt(10));
egf := -2*exp(x) + (1-s)*exp(-sqrt(10)*x) + (1+s)*exp(sqrt(10)*x);
ser := series(egf, x, len+2): seq(simplify(n!*coeff(ser,x,n)), n = 1..len) end:
A050250List(25); # Peter Luschny, Jun 11 2022 after Stefano Spezia

LinearRecurrence[{1,10,-10},{9,18,108},30] (* Harvey P. Dale, Jan 29 2012 *)
CoefficientList[Series[2Cosh[Sqrt[10]x]-2(Cosh[x]+Sinh[x])+11Sinh[Sqrt[10]x]/Sqrt[10],{x,0,25}],x]Table[n!,{n,0,25}] (* Stefano Spezia, Jun 11 2022 *)

a(n)=10^(n\2)*(13-9*(-1)^n)/2-2 \\ Charles R Greathouse IV, Jun 25 2017

def a(n):
  m = 10 ** (n >> 1)
  if n & 1 == 0:
    return (m - 1) << 1
  else:
    return (11 * m) - 2 # Darío Clavijo, Oct 16 2023

a(2*k) = 2*10^k - 2, a(2*k + 1) = 11*10^k - 2. - Sascha Kurz, Apr 14 2002
From Jonathan Vos Post, Jun 18 2008: (Start)
a(n) = Sum_{i=1..n} A050683(i).
a(n) = Sum_{i=1..n} 9*10^floor((i-1)/2).
a(n) = 9*Sum_{i=1..n} 10^floor((i-1)/2). (End)
From Bruno Berselli, Feb 15 2011: (Start)
G.f.: 9*x*(1+x)/((1-x)*(1-10*x^2)).
a(n) = (1/2)*10^((2*n + (-1)^n - 1)/4)*(13 - 9*(-1)^n) - 2. (End)
a(1)=9, a(2)=18, a(3)=108; for n>3, a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3). - Harvey P. Dale, Jan 29 2012
a(n) = 10*a(n-2) + 18. - R. J. Mathar, Nov 07 2015
E.g.f.: 2*cosh(sqrt(10)*x) - 2*(cosh(x) + sinh(x)) + 11*sinh(sqrt(10)*x)/sqrt(10). - Stefano Spezia, Jun 11 2022

More terms from Patrick De Geest, Dec 15 1999

a(24)-a(25) from Jonathan Vos Post, Jun 18 2008

A118594 Palindromes in base 3 (written in base 3).

Original entry on oeis.org

0, 1, 2, 11, 22, 101, 111, 121, 202, 212, 222, 1001, 1111, 1221, 2002, 2112, 2222, 10001, 10101, 10201, 11011, 11111, 11211, 12021, 12121, 12221, 20002, 20102, 20202, 21012, 21112, 21212, 22022, 22122, 22222, 100001, 101101, 102201, 110011, 111111, 112211, 120021

Offset: 1

Martin Renner, May 08 2006

The number of n-digit terms is given by A225367. - M. F. Hasler, May 05 2013 [Moved here on May 08 2013]
Digit-wise application of A000578 (and also superposition of a(n) with its horizontal OR vertical reflection) yields A006072. - M. F. Hasler, May 08 2013
Equivalently, palindromes k (written in base 10) such that 4*k is a palindrome. - Bruno Berselli, Sep 12 2018

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Ternary

Cf. A007089, A014190, A057148, A118595, A118596, A118597, A118598, A118599, A118600, A002113.

(* get NextPalindrome from A029965 *) Select[NestList[NextPalindrome, 0, 1110], Max@IntegerDigits@# < 3 &] (* Robert G. Wilson v, May 09 2006 *)
Select[FromDigits/@Tuples[{0,1,2},8],IntegerDigits[#]==Reverse[ IntegerDigits[ #]]&] (* Harvey P. Dale, Apr 20 2015 *)

{for(l=1,5,u=vector((l+1)\2,i,10^(i-1)+(2*i-11&&i==1,2]), print1(v*u",")))} \\ The n-th term could be produced by using (partial sums of) A225367 to skip all shorter terms, and then skipping the adequate number of vectors v until n is reached.  - M. F. Hasler, May 08 2013

from itertools import count, islice, product
def agen(): # generator of terms
    yield from [0, 1, 2]
    for d in count(2):
        for start in "12":
            for rest in product("012", repeat=d//2-1):
                left = start + "".join(rest)
                for mid in [[""], ["0", "1", "2"]][d%2]:
                    yield int(left + mid + left[::-1])
print(list(islice(agen(), 42))) # Michael S. Branicky, Mar 29 2022

from sympy import integer_log
from gmpy2 import digits
def A118594(n):
    if n == 1: return 0
    y = 3*(x:=3**integer_log(n>>1,3)[0])
    return int((s:=digits(n-x,3))+s[-2::-1] if nChai Wah Wu, Jun 14 2024

[int(n.str(base=3)) for n in (0..757) if Word(n.digits(3)).is_palindrome()] # Peter Luschny, Sep 13 2018

More terms from Robert G. Wilson v, May 09 2006

a(40) and beyond from Michael S. Branicky, Mar 29 2022

A134810 Giza numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 121, 232, 343, 454, 565, 676, 787, 898, 12321, 23432, 34543, 45654, 56765, 67876, 78987, 1234321, 2345432, 3456543, 4567654, 5678765, 6789876, 123454321, 234565432, 345676543, 456787654, 567898765, 12345654321, 23456765432

Offset: 1

Omar E. Pol, Nov 25 2007, Nov 26 2007

For n > 9 the structure of digits represents the pyramids of Giza. Also the top of a mountain. The first digit is equal to the last digit. The first digits are in consecutive increasing order. The last digits are in consecutive decreasing order. The largest digit is the central digit. The number of digits is odd. This sequence has 45 terms. The final term is 12345678987654321. Giza numbers are mountain numbers A134941 and palindromes A002113.

There are 10 - k numbers with 2*k - 1 digits. - Omar E. Pol, Aug 04 2011

			Illustration using the final term of this sequence:
  . . . . . . . . 9 . . . . . . . .
  . . . . . . . 8 . 8 . . . . . . .
  . . . . . . 7 . . . 7 . . . . . .
  . . . . . 6 . . . . . 6 . . . . .
  . . . . 5 . . . . . . . 5 . . . .
  . . . 4 . . . . . . . . . 4 . . .
  . . 3 . . . . . . . . . . . 3 . .
  . 2 . . . . . . . . . . . . . 2 .
  1 . . . . . . . . . . . . . . . 1

Nathaniel Johnston, Table of n, a(n) for n = 1..45 (full sequence)

Cf. A002113, A134941.

ups = Flatten[Table[Range[i, j - 1], {i, 1, 9}, {j, i + 1, 10}], 1];afull = Sort[  Map[ToExpression@StringJoin@Map[ToString, #[[;; -2]] ~Join~ Reverse[#]] &, ups]];afull (* James C. McMahon, Apr 11 2025 *)

ups = [tuple(range(i, j)) for i in range(1, 10) for j in range(i+1, 11)]
afull = sorted(int("".join(map(str, u[:-1] + u[::-1]))) for u in ups)
print(afull) # Michael S. Branicky, Aug 02 2022

A178333(a(n))*A136522(a(n)) = 1. - Reinhard Zumkeller, May 25 2010

A181965 a(n) = 10^(2n+1) - 10^n - 1.

Original entry on oeis.org

8, 989, 99899, 9998999, 999989999, 99999899999, 9999998999999, 999999989999999, 99999999899999999, 9999999998999999999, 999999999989999999999, 99999999999899999999999, 9999999999998999999999999, 999999999999989999999999999, 99999999999999899999999999999, 9999999999999998999999999999999

Offset: 0

Ivan Panchenko, Apr 04 2012

n 9's followed by an 8 followed by n 9's.

See A183187 = {26, 378, 1246, 1798, 2917, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)8(9), updated: June 25, 2017.
Makoto Kamada, Factorization of 99...99899...99, updated Dec 11 2018.
Markus Tervooren, Factorizations of (9)w8(9)w, FactorDB.com
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077794-1)/2 = A183187 (indices of primes).
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332190 .. A332197 (variants with different middle digit 0, ..., 7).

A181965 := n -> 10^(2*n+1)-1-10^n; # M. F. Hasler, Feb 08 2020

Array[10^(2 # + 1) - 1- 10^# &, 15, 0] (*  M. F. Hasler, Feb 08 2020 *)
Table[With[{c=PadRight[{},n,9]},FromDigits[Join[c,{8},c]]],{n,0,20}] (* Harvey P. Dale, Jun 07 2021 *)

apply( {A181965(n)=10^(n*2+1)-1-10^n}, [0..15]) \\ M. F. Hasler, Feb 08 2020

def A181965(n): return 10**(n*2+1)-1-10^n # M. F. Hasler, Feb 08 2020

From M. F. Hasler, Feb 08 2020: (Start)
a(n) = 9*A138148(n) + 8*10^n = A002283(2n+1) - A011557(10^n).
G.f.: (8 + 101*x - 1000*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. (End)

Edited and extended to a(0) = 8 by M. F. Hasler, Feb 10 2020

A259381 Palindromic numbers in bases 3 and 8 written in base 10.

Original entry on oeis.org

0, 1, 2, 4, 121, 130, 203, 316, 8578, 9490, 17492, 944035, 1141652, 1276916, 1554173, 58961443, 67470916, 4099065139, 5691134677, 81452592329, 81473867465, 419572845958, 21056462595764, 363376288168081

Offset: 1

Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015

			121 is in the sequence because 121_10 = 171_8 = 11111_3.

Giovanni Resta, Table of n, a(n) for n = 1..43
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, 8]; If[palQ[pp, 3], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=3; b2=8; 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 A014190 and A029803.

A259383 Palindromic numbers in bases 5 and 8 written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 18, 36, 186, 438, 2268, 2709, 11898, 18076, 151596, 228222, 563786, 5359842, 32285433, 257161401, 551366532, 621319212, 716064597, 2459962002, 5018349804, 5067084204, 7300948726, 42360367356, 139853034114, 176616961826, 469606524278, 669367713609, 1274936571666, 1284108810066, 5809320306961, 8866678870082, 11073162740322, 14952142559323, 325005646077513

Offset: 1

Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015

			186 is in the sequence because 186_10 = 272_8 = 1221_5.

Giovanni Resta, Table of n, a(n) for n = 1..67
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, 8]; If[palQ[pp, 5], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=5; b2=8; 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 A029803.

A259387 Palindromic numbers in bases 4 and 9 written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 255, 273, 373, 546, 2550, 2730, 2910, 16319, 23205, 54215, 1181729, 1898445, 2576758, 3027758, 3080174, 4210945, 9971750, 163490790, 2299011170, 6852736153, 6899910553, 160142137430, 174913133450, 204283593150, 902465909895, 1014966912315, 2292918574418, 9295288254930, 11356994802010, 11372760382810, 38244097345762

Offset: 1

Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015

			273 is in the sequence because 273_10 = 333_9 = 10101_4.

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, 4], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=4; 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 A014192 and A029955.

cp's OEIS Frontend

A259390 Palindromic numbers in bases 7 and 9 written in base 10.

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A332120 a(n) = 2*(10^(2n+1)-1)/9 - 2*10^n.

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A332190 a(n) = 10^(2n+1) - 1 - 9*10^n.

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A050250 Number of nonzero palindromes less than 10^n.

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A118594 Palindromes in base 3 (written in base 3).

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A134810 Giza numbers.

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A181965 a(n) = 10^(2n+1) - 10^n - 1.

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A332120 a(n) = 2(10^(2n+1)-1)/9 - 210^n.