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a(n) = 13*n + 2.
9

%I #42 Jan 17 2024 09:08:45

%S 2,15,28,41,54,67,80,93,106,119,132,145,158,171,184,197,210,223,236,

%T 249,262,275,288,301,314,327,340,353,366,379,392,405,418,431,444,457,

%U 470,483,496,509,522,535,548,561,574,587,600,613,626,639,652,665,678,691

%N a(n) = 13*n + 2.

%C Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 2, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no term is a cube. - _Bruno Berselli_, Feb 19 2016

%C Numbers k such that GCD(2*k^5+1, 3*k^3+2) > 1. This GCD is 13 if k == 2 (mod 13), or 1 otherwise. - _Philippe Deléham_, Jan 16 2024

%H Vincenzo Librandi, <a href="/A153080/b153080.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F G.f.: (2+11*x)/(1-x)^2. - _R. J. Mathar_, Jan 05 2011

%F a(n) = 2*a(n-1) - a(n-2). - _Vincenzo Librandi_, Feb 25 2012

%p A153080:=n->13*n+2: seq(A153080(n), n=0..100); # _Wesley Ivan Hurt_, Oct 05 2017

%t Range[2, 1000, 13] (* _Vladimir Joseph Stephan Orlovsky_, May 29 2011 *)

%t LinearRecurrence[{2,-1},{2,15},30] (* _Vincenzo Librandi_, Feb 25 2012 *)

%o (Magma) I:=[2, 15]; [n le 2 select I[n] else 2*Self(n-1)-1*Self(n-2): n in [1..60]]; // _Vincenzo Librandi_, Feb 25 2012

%Y Cf. A008595, A141858, A190991.

%Y Cf. A269100. - _Bruno Berselli_, Feb 22 2016

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Feb 10 2009