A070199 Number of palindromes of length <= n.
10, 19, 109, 199, 1099, 1999, 10999, 19999, 109999, 199999, 1099999, 1999999, 10999999, 19999999, 109999999, 199999999, 1099999999, 1999999999, 10999999999, 19999999999, 109999999999, 199999999999, 1099999999999, 1999999999999, 10999999999999, 19999999999999
Offset: 1
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (1,10,-10).
Programs
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Mathematica
LinearRecurrence[{1,10,-10},{10,19,109},30] (* Harvey P. Dale, Mar 18 2016 *) -
PARI
Vec(x*(10+9*x-10*x^2)/((1-x)*(1-10*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017
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Python
def A070199(n): return 10**(n>>1)*(11 if n&1 else 2)-1 # Chai Wah Wu, Jul 30 2025
Formula
From Colin Barker, Jun 30 2012: (Start)
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3).
G.f.: x*(10 + 9*x - 10*x^2)/((1 - x)*(1 - 10*x^2)). (End)
a(n) = (-2*sqrt(10)+10^(n/2)*(11+2*sqrt(10)+(-1)^n*(-11+2*sqrt(10))))/(2*sqrt(10)). - Harvey P. Dale, Mar 18 2016
From Colin Barker, Mar 17 2017: (Start)
a(n) = 2^(n/2 + 1)*5^(n/2) - 1 for n even.
a(n) = 11*10^((n-1)/2) - 1 for n odd. (End)
a(n) = A050250(n) + 1. - Andrew Howroyd, Oct 28 2020
E.g.f.: 2*cosh(sqrt(10)*x) - cosh(x) - 1 - sinh(x) + 11*sinh(sqrt(10)*x)/sqrt(10). - Stefano Spezia, Jul 01 2023