A225810 a(n) = (10^n)^2 + 4*(10^n) + 1.

6, 141, 10401, 1004001, 100040001, 10000400001, 1000004000001, 100000040000001, 10000000400000001, 1000000004000000001, 100000000040000000001, 10000000000400000000001, 1000000000004000000000001, 100000000000040000000000001, 10000000000000400000000000001

Offset: 0

Author

Lance J. Weingartz, Jul 29 2013

Keywords

• nonn
• easy

Comments

This is an instance of (10^n)^2 + x(10^n) + 1 which umbrellas A066138, A033934, A171375, A171410, A171461, A171513 and A171553 which all produce palindromes of the form 1...n...1 when n <> 0.

Links

• Colin Barker, Table of n, a(n) for n = 0..499
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A033934, A066138, A171375, A171410, A171461, A171513, A171553.

Programs

Mathematica

Table[(10^n)^2 + 4*(10^n) + 1, {n, 0, 20}] (* T. D. Noe, Aug 12 2013 *)
LinearRecurrence[{111,-1110,1000},{6,141,10401},20] (* Harvey P. Dale, Oct 28 2017 *)

PARI

Vec(-3*(470*x^2-175*x+2)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100)) \\ Colin Barker, Apr 27 2015

Formula

From Colin Barker, Apr 27 2015: (Start)

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).

G.f.: -3*(470*x^2-175*x+2)/((x-1)*(10*x-1)*(100*x-1)). (End)

E.g.f.: exp(x)*(1 + 4*exp(9*x) + exp(99*x)). - Elmo R. Oliveira, Jul 04 2025