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%I #54 Mar 06 2023 10:06:03
%S 1,5,32,145,512,1649,5392,18785,69632,268705,1058576,4208945,16797952,
%T 67137425,268473872,1073792449,4295032832,17179952705,68719581712,
%U 274878037265,1099511787776,4398046705585,17592186278672
%N a(n) = 4^n + n^4.
%C a(n) is prime if and only if n = 1. - _Reinhard Zumkeller_, May 24 2009
%C The statement above (and the corollary that 5 is the only prime term in this sequence) can be proved using Sophie Germain's identity x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). - _Alonso del Arte_, Oct 31 2013 [exponents corrected by _Thomas Ordowski_, Nov 29 2017]
%H Vincenzo Librandi, <a href="/A001589/b001589.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="http://dx.doi.org/10.1007/BF03023374">Notes</a>, Mathematical Intelligencer 2(2) (1980), p. 66. - _Reinhard Zumkeller_, May 24 2009
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (9,-30,50,-45,21,-4).
%F G.f.: -(5*x^5 + 38*x^4 + 43*x^3 - 17*x^2 + 4*x - 1) / ((x - 1)^5*(4*x - 1)). - _Colin Barker_, Jan 01 2013
%t Table[4^n + n^4, {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 14 2011 *)
%t LinearRecurrence[{9,-30,50,-45,21,-4},{1,5,32,145,512,1649},30] (* _Harvey P. Dale_, Mar 06 2023 *)
%o (PARI) a(n)=1<<(n+n)+n^4
%o (Magma) [4^n+n^4: n in [0..30]]; // _Vincenzo Librandi_, Oct 27 2011
%Y Cf. A001580, A001585.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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