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a(n) = n^2 + 13.
2

%I #21 Sep 08 2022 08:46:07

%S 13,14,17,22,29,38,49,62,77,94,113,134,157,182,209,238,269,302,337,

%T 374,413,454,497,542,589,638,689,742,797,854,913,974,1037,1102,1169,

%U 1238,1309,1382,1457,1534,1613,1694,1777,1862,1949,2038,2129,2222,2317

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

%C For i=0..28, 2*a(i)+3 is prime . - _Vincenzo Librandi_, Jun 01 2014

%H Vincenzo Librandi, <a href="/A241749/b241749.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (13 - 25*x + 14*x^2)/(1 - x)^3.

%F a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.

%F From _Amiram Eldar_, Nov 02 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + sqrt(13)*Pi*coth(sqrt(13)*Pi))/26.

%F Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(13)*Pi*cosech(sqrt(13)*Pi))/26. (End)

%t Table[n^2 + 13, {n, 0, 60}]

%o (Magma) [n^2+13: n in [0..60]];

%o (PARI) a(n)=n^2+13 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. similar sequences listed in A114962.

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Apr 30 2014