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

1,1

Comments

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

Links

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).

Formula

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

Maple

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

Mathematica

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

Prog

(PARI) A050683(n)=9*10^((n-1)\2) \\ M. F. Hasler, Nov 16 2008
(PARI) \\ 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
(PARI) 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
(PARI) a(n) = if(n<3, 9, 10*a(n-2)); \\ Altug Alkan, Oct 03 2018
(Magma) [9*10^Floor((n-1)/2): n in [1..30]]; // Vincenzo Librandi, Aug 16 2011
(GAP) a:=[9, 9];; for n in [3..30] do a[n]:=10*a[n-2]; od; a; # Muniru A Asiru, Oct 07 2018

Crossrefs

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.

Sequence in context: A323210 A215272 A165427 * A210095 A092548 A121389

Adjacent sequences: A050680 A050681 A050682 * A050684 A050685 A050686

Keyword

nonn,easy,base,nice

Author

Patrick De Geest, Aug 15 1999

Status

approved