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a(n) = 10^(2*n+2) + 111*10^n + 1.
1

%I #11 May 24 2021 01:02:19

%S 212,11111,1011101,100111001,10001110001,1000011100001,

%T 100000111000001,10000001110000001,1000000011100000001,

%U 100000000111000000001,10000000001110000000001,1000000000011100000000001,100000000000111000000000001,10000000000001110000000000001

%N a(n) = 10^(2*n+2) + 111*10^n + 1.

%C For n > 1, palindromic numbers of the form 10..01110..01.

%C This is the earliest sequence of the form 10^(2*n+t) + A002275(t+1)*10^n + 1 that contains primes of the form mentioned in the previous comment. For example, the terms of the sequence for t = 0 are all divisible by 3 (see A066138, where 3 is the only prime), while each term b(i) of the sequence with t = 1 (A319667) is divisible by 10^i+1.

%C For the values of n such that a(n) is prime, see A344424.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).

%F G.f.: -(13100*x^2 - 12421*x + 212)/(1000*x^3 - 1110*x^2 + 111*x - 1). - _Jinyuan Wang_, May 22 2021

%F a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3). - _Wesley Ivan Hurt_, May 22 2021

%F E.g.f.: exp(x)*(1 + 111*exp(9*x) + 100*exp(99*x)). - _Stefano Spezia_, May 22 2021

%o (PARI) a(n) = 10^(2*n+2) + 111*10^n + 1

%Y Cf. A002275, A066138, A319667, A344424.

%K nonn,easy

%O 0,1

%A _Felix Fröhlich_, May 18 2021