A070199 -id:A070199 - cp's OEIS frontend

A050683 Number of nonzero palindromes of length n.

9, 9, 90, 90, 900, 900, 9000, 9000, 90000, 90000, 900000, 900000, 9000000, 9000000, 90000000, 90000000, 900000000, 900000000, 9000000000, 9000000000, 90000000000, 90000000000, 900000000000, 900000000000, 9000000000000

Offset: 1

Patrick De Geest, Aug 15 1999

In general the number of base k palindromes with n digits is (k-1)*k^floor((n-1)/2). (See A117855 or A225367 for an explanation.) - Henry Bottomley, Aug 14 2000

This sequence does not count 0 as palindrome with 1 digit, see A070252 = (10,9,90,90,...) for the variant which does. - M. F. Hasler, Nov 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dr. Math, More info 1.
Dr. Math, More info 2.
Index entries for linear recurrences with constant coefficients, signature (0,10).

Cf. A002113, A050250, A050251, A070252, A070199.
Cf. A016116 for numbers of binary palindromes, A016115 for prime palindromes.
Cf. A117855 for the base 3 version, and A225367 for a variant.

a:=[9,9];; for n in [3..30] do a[n]:=10*a[n-2]; od; a; # Muniru A Asiru, Oct 07 2018

[9*10^Floor((n-1)/2): n in [1..30]]; // Vincenzo Librandi, Aug 16 2011

seq(9*10^floor((n-1)/2),n=1..30); # Muniru A Asiru, Oct 07 2018

With[{c=9*10^Range[0,20]},Riffle[c,c]] (* or *) LinearRecurrence[{0,10},{9,9},40] (* Harvey P. Dale, Dec 15 2013 *)

A050683(n)=9*10^((n-1)\2) \\ M. F. Hasler, Nov 16 2008

\\ using M. F. Hasler's is_A002113(n) from A002113
is_A002113(n)={Vecrev(n=digits(n))==n}
for(n=1,8,j=0;for(k=10^(n-1),10^n-1,if(is_A002113(k),j++));print1(j,", ")) \\ Hugo Pfoertner, Oct 03 2018

is_palindrome(x)={my(d=digits(x));for(k=1,#d\2,if(d[k]!=d[#d+1-k],return(0)));return(1)}
for(n=1,8,j=0;for(k=10^(n-1),10^n-1,if(is_palindrome(k),j++));print1(j,", ")) \\ Hugo Pfoertner, Oct 02 2018

a(n) = if(n<3, 9, 10*a(n-2)); \\ Altug Alkan, Oct 03 2018

def A050683(n): return 9*10**(n-1>>1) # Chai Wah Wu, Jul 30 2025

a(n) = 9*10^floor((n-1)/2).
From Colin Barker, Apr 06 2012: (Start)
a(n) = 10*a(n-2).
G.f.: 9*x*(1+x)/(1-10*x^2). (End)
E.g.f.: 9*(cosh(sqrt(10)*x) + sqrt(10)*sinh(sqrt(10)*x) - 1)/10. - Stefano Spezia, Jun 11 2022

A070252 Number of n-digit palindromes.

Original entry on oeis.org

10, 9, 90, 90, 900, 900, 9000, 9000, 90000, 90000, 900000, 900000, 9000000, 9000000, 90000000, 90000000, 900000000, 900000000, 9000000000, 9000000000, 90000000000, 90000000000, 900000000000, 900000000000, 9000000000000

Offset: 1

Amarnath Murthy, May 06 2002

Index entries for linear recurrences with constant coefficients, signature (0,10).

A variant of A050683, which is the principal entry for this sequence. Cf. A016115.

Partial sums give A070199.

LinearRecurrence[{0,10},{10,9,90},25] (* Stefano Spezia, Jun 11 2022 *)

def A070252(n): return 10 if n==1 else 9*10**(n-1>>1) # Chai Wah Wu, Jul 30 2025

From Colin Barker, Aug 19 2013: (Start)
a(n) = 10*a(n-2) for n>3.
G.f.: x*(10*x^2-9*x-10) / (10*x^2-1). (End)
E.g.f.: x + 9*(cosh(sqrt(10)*x) - 1 + sqrt(10)*sinh(sqrt(10)*x))/10. - Stefano Spezia, Jun 11 2022

A050251 Number of palindromic primes less than 10^n.

Original entry on oeis.org

0, 4, 5, 20, 20, 113, 113, 781, 781, 5953, 5953, 47995, 47995, 401696, 401696, 3438339, 3438339, 30483565, 30483565, 269577430, 269577430, 2427668363, 2427668363, 22170468927, 22170468927, 202985860292, 202985860292

Offset: 0

Eric W. Weisstein

Every palindrome with an even number of digits is divisible by 11 and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits, 11. - Martin Renner, Apr 15 2006

Patrick De Geest, World!Of Palindromic Primes, Page 1
Shyam Sunder Gupta, Palindromic Primes up to 10^19.
Shyam Sunder Gupta, Palindromic Primes up to 10^23.
Shyam Sunder Gupta, Palindromic Primes up to 10^25.
Eric Weisstein's World of Mathematics, Palindromic Prime.
Index entries for sequences related to numbers of primes in various ranges

Partial sums of A016115.

Cf. A002113 (palindromes), A002385 (palindromic primes).

from _future_ import division
from sympy import isprime
def paloddgen(l,b=10): # generator of odd-length palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            n = b**(x-1)
            n2 = n*b
            for y in range(n,n2):
                k, m = y//b, 0
                while k >= b:
                    k, r = divmod(k,b)
                    m = b*m + r
                yield y*n + b*m + k
def A050251(n):
    if n <= 1:
        return 4*n
    else:
        c = 1
        for i in paloddgen((n+1)//2):
            if isprime(i):
                c += 1
        return c # Chai Wah Wu, Jan 05 2015

a(n) ~ A070199(n)/log(10^n) = 1/log(10^n)*Sum {k=1..n} 9*10^floor[(k-1)/2]. - Robert G. Wilson v, May 31 2009

a(2n) = a(2n-1) for n > 1. - Chai Wah Wu, Nov 21 2021

More terms from Patrick De Geest, Aug 01 1999
2 more terms from Shyam Sunder Gupta, Feb 12 2006
2 more terms from Shyam Sunder Gupta, Mar 13 2009
a(23)-a(24) from Shyam Sunder Gupta, Oct 05 2013
Missing a(0) inserted by Chai Wah Wu, Nov 21 2021
a(25)-a(26) from Shyam Sunder Gupta, Dec 19 2024

A328332 Expansion of (1 + 4x - 5x^2 + 10x^3) / ((1 - x) (1 - 10*x^2)).

Original entry on oeis.org

1, 5, 10, 60, 110, 610, 1110, 6110, 11110, 61110, 111110, 611110, 1111110, 6111110, 11111110, 61111110, 111111110, 611111110, 1111111110, 6111111110, 11111111110, 61111111110, 111111111110, 611111111110, 1111111111110, 6111111111110, 11111111111110, 61111111111110, 111111111111110

Offset: 0

Ilya Gutkovskiy, Oct 12 2019

Number of odd palindromes <= 10^n.

Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A002113, A029950, A050250, A070199, A328333.

nmax = 28; CoefficientList[Series[(1 + 4 x - 5 x^2 + 10 x^3) / ((1 - x) (1 - 10 x^2)), {x, 0, nmax}], x]
Join[{1}, LinearRecurrence[{1, 10, -10}, {5, 10, 60}, 28]]

Vec((1 + 4*x - 5*x^2 + 10*x^3) / ((1 - x) * (1 - 10*x^2)) + O(x^30)) \\ Michel Marcus, Oct 13 2019

G.f.: (1 + 4*x - 5*x^2 + 10*x^3) / ((1 - x) * (1 - 10*x^2)).

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3). - Wesley Ivan Hurt, Aug 25 2022

A328333 Expansion of (1 + 4x - 6x^2) / ((1 - x) * (1 - 10*x^2)).

Original entry on oeis.org

1, 5, 9, 49, 89, 489, 889, 4889, 8889, 48889, 88889, 488889, 888889, 4888889, 8888889, 48888889, 88888889, 488888889, 888888889, 4888888889, 8888888889, 48888888889, 88888888889, 488888888889, 888888888889, 4888888888889, 8888888888889, 48888888888889, 88888888888889

Offset: 0

Ilya Gutkovskiy, Oct 12 2019

Number of even palindromes < 10^n.

Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A002113, A029951, A050250, A070199, A328332.

nmax = 28; CoefficientList[Series[(1 + 4 x - 6 x^2)/((1 - x) (1 - 10 x^2)), {x, 0, nmax}], x]
LinearRecurrence[{1, 10, -10}, {1, 5, 9}, 29]

Vec((1 + 4*x - 6*x^2) / ((1 - x) * (1 - 10*x^2)) + O(x^30)) \\ Michel Marcus, Oct 13 2019

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A050683 Number of nonzero palindromes of length n.

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A070252 Number of n-digit palindromes.

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A050251 Number of palindromic primes less than 10^n.

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A328332 Expansion of (1 + 4*x - 5*x^2 + 10*x^3) / ((1 - x) * (1 - 10*x^2)).

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A328333 Expansion of (1 + 4*x - 6*x^2) / ((1 - x) * (1 - 10*x^2)).

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Mathematica

PARI

A328332 Expansion of (1 + 4x - 5x^2 + 10x^3) / ((1 - x) (1 - 10*x^2)).

A328333 Expansion of (1 + 4x - 6x^2) / ((1 - x) * (1 - 10*x^2)).