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

1,1

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

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

Mathematica

LinearRecurrence[{1, 10, -10}, {10, 19, 109}, 30] (* Harvey P. Dale, Mar 18 2016 *)

Prog

(PARI) Vec(x*(10+9*x-10*x^2)/((1-x)*(1-10*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

Crossrefs

Partial sums of A070252.

Cf. A050250.

Sequence in context: A166706 A131495 A060630 * A015445 A220005 A253213

Adjacent sequences: A070196 A070197 A070198 * A070200 A070201 A070202

Keyword

nonn,base,easy

Author

N. J. A. Sloane and Robert G. Wilson v, May 14 2002

Status

approved