A240022 Total number of digits in palindromes with n digits.

10, 18, 270, 360, 4500, 5400, 63000, 72000, 810000, 900000, 9900000, 10800000, 117000000, 126000000, 1350000000, 1440000000, 15300000000, 16200000000, 171000000000, 180000000000, 1890000000000, 1980000000000, 20700000000000, 21600000000000

Offset

1,1

Comments

Let f(1) = g(1) = 10 and f(2) = 1; d(n) denotes the number of digits in f(n) and for n >= 3, f(n) = 10*f(n-1) + 5*10^(d(n-1)-1) if n is odd, otherwise f(n) = f(n-1) + 10^(d(n-1)-1)/2. Let g(n) = 18*f(n) for n > 1. It gives g(2) = 18, g(3) = 270, g(4) = 360, g(5) = 4500, .... In fact g(n) produces a different sequence than a(n).

Links

Table of n, a(n) for n=1..24.

Index entries for sequences related to palindromes

Index entries for linear recurrences with constant coefficients, signature (0,20,0,-100).

Formula

a(n) = n*A070252(n).

a(n) = 20*a(n-2)-100*a(n-4) for n>5. G.f.: 2*x*(50*x^4+35*x^2+9*x+5) / (10*x^2-1)^2. - Colin Barker, Mar 31 2014

Example

There are nine 2-digit palindromes, so a(2) = 2*9 = 18.

Prog

(PARI) print1("10, 18, "); m=9; for(n=3, 24, if(bitand(n, 1), m=10*m); print1(m*n, ", "));
(PARI) Vec(2*x*(50*x^4+35*x^2+9*x+5)/(10*x^2-1)^2 + O(x^100)) \\ Colin Barker, Mar 31 2014

Crossrefs

Cf. A008600, A070252.

Sequence in context: A358985 A171767 A153689 * A264386 A173822 A122607

Adjacent sequences: A240019 A240020 A240021 * A240023 A240024 A240025

Keyword

nonn,base,easy

Author

Arkadiusz Wesolowski, Mar 30 2014

Status

approved