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%I #76 Oct 20 2024 13:01:17
%S 29,32,37,44,53,64,77,92,109,128,149,172,197,224,253,284,317,352,389,
%T 428,469,512,557,604,653,704,757,812,869,928,989,1052,1117,1184,1253,
%U 1324,1397,1472,1549,1628,1709,1792,1877,1964,2053,2144,2237,2332,2429,2528
%N a(n) = n^2 + 2*n + 29.
%C A139851, which lists primes of the form 4x^2 + 4xy + 29y^2, contains all prime values of a(n). - _Altug Alkan_, Oct 02 2015
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = a(n-1) + A005408(n), a(0) = 29, for n > 0. - _Altug Alkan_, Oct 02 2015
%F From _Vincenzo Librandi_, Oct 03 2015: (Start)
%F G.f.: (29 - 55*x + 28*x^2)/(1-x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F a(n) = A005563(n) + 29. - _Omar E. Pol_, Oct 17 2015
%F E.g.f.: (29 + 3*x + x^2)*exp(x). - _Elmo R. Oliveira_, Oct 19 2024
%e For n = 3, a(3) = 3^2 + 2*3 + 29 = 44.
%t Table[n^2 + 2 n + 29, {n, 0, 50}] (* _Bruno Berselli_, Oct 25 2015 *)
%t LinearRecurrence[{3,-3,1},{29,32,37},50] (* _Harvey P. Dale_, Oct 14 2023 *)
%o (Python) def a(x):return x*x+2*x+27
%o (PARI) vector(50, n, n--; n^2+2*n+29) \\ _Altug Alkan_, Oct 02 2015
%o (PARI) Vec((29 - 55*x + 28*x^2)/(1-x)^3 + O(x^100)) \\ _Altug Alkan_, Oct 17 2015
%o (Magma) [n^2+2*n+29: n in [0..50]]; // _Vincenzo Librandi_, Oct 03 2015
%Y Cf. A005408, A005563, A139851.
%K nonn,easy
%O 0,1
%A _Jake Saville_, Oct 02 2015