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%I #37 Feb 24 2023 04:29:00
%S 2,6,30,110,306,702,1406,2550,4290,6806,10302,15006,21170,29070,39006,
%T 51302,66306,84390,105950,131406,161202,195806,235710,281430,333506,
%U 392502,459006,533630,617010,709806,812702,926406,1051650,1189190,1339806,1504302,1683506
%N a(n) = (n^2 + 1)*(n^2 + 2).
%C a(n) and A304578(n) are coprime for all n.
%H Daniele Mastrostefano and Carlo Sanna, <a href="https://arxiv.org/abs/1805.05114">On numbers n with polynomial image coprime with the nth term of a linear recurrence</a>, arXiv:1805.05114. [math.NT], 2018 (see 4.2, page 7).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: 2*(1 - 2*x + 10*x^2 + 3*x^4)/(1 - x)^5.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F a(n) = A002378(A002522(n)). - _Altug Alkan_, May 17 2018
%F Sum_{n>=0} 1/a(n) = 1/4 + coth(Pi)*Pi/2 - coth(sqrt(2)*Pi)*Pi/(2*sqrt(2)). - _Amiram Eldar_, Feb 24 2023
%t CoefficientList[Series[2 (1 - 2 x + 10 x^2 + 3 x^4) / (1 - x)^5, {x, 0, 35}], x] (* or *) Table[(n^2 + 1) (n^2 + 2), {n, 0, 40}]
%t LinearRecurrence[{5,-10,10,-5,1},{2,6,30,110,306},40] (* _Harvey P. Dale_, Nov 13 2022 *)
%o (Magma) [(n^2+1)*(n^2+2): n in [0..40]];
%o (PARI) a(n) = my(k=n^2+1); k*(k+1); \\ _Altug Alkan_, May 17 2018
%Y Cf. A002522, A304578.
%Y Subsequence of A002378, A045619, A279019.
%K nonn,easy
%O 0,1
%A _Vincenzo Librandi_, May 17 2018
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