A050250 Number of nonzero palindromes less than 10^n.

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

Author

Eric W. Weisstein, Dec 11 1999

Keywords

• nonn
• easy
• base
• nice

Links

• 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).

Crossrefs

Cf. A002113, A002778, A050683.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

More terms from Patrick De Geest, Dec 15 1999

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