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%I #46 Sep 08 2022 08:45:46
%S 858,3139,13142,36807,80098,149003,249534,387727,569642,801363,
%T 1088998,1438679,1856562,2348827,2921678,3581343,4334074,5186147,
%U 6143862,7213543,8401538,9714219,11157982,12739247,14464458,16340083,18372614,20568567,22934482,25476923
%N a(n) = n^4 + 984*n^3 + 902*n^2 + 394*n + 858.
%C Comment (entirely taken from Cugiani's text - see References) from _Vincenco Librandi_, Aug 23 2011: (Start)
%C This deals with primitive polynomials in GF_k(p). There are p^k monic k-th order polynomials J(p) = x^k + a(k-1)*x^(k-1) + ... + a(0), because there are k independent coefficients a(.), each restricted modulo the prime p. phi(p^k-1)/k of these polynomials are primitive, where phi=A000010. [Example for p=7 and k=2: phi(7^2-1)/2 = phi(48)/2 = 16/2=8. See A011260 for p=2, A027385 for p=3, A027741 for p=5 etc.] Of these sets of primitive polynomials we select with p=1031 the polynomial x^3+73*x^2+x+67 for k=3 in A163303 and x^4+984*x^3+90*x^2+394-x+858 for k=4 in A163304 by the following criteria (This could be extended to k=5, 6,...): Let r = (p^k -1)/(p-1). We demand (see Theorem 1 in Hansen-Mullen)
%C i) (-1)^k a(0) is a primitive element of J(p).
%C ii) The remainder of the division of x^r through the polynomial equals (-1)^k a(0).
%C iii) The remainder of the division of x^(r/q) through the polynomial must have positive degree for each prime divisor q|r.
%C (End)
%D Marco Cugiani, Metodi numerico statistici (Collezione di Matematica applicata n.7), UTET Torino, 1980, pp.78-84
%H Vincenzo Librandi, <a href="/A163304/b163304.txt">Table of n, a(n) for n = 0..10000</a>
%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.: (858-1151*x+6027*x^2-6093*x^3+383*x^4)/(1-x)^5. - _Bruno Berselli_, Aug 24 2011
%F From _G. C. Greubel_, Dec 18 2016: (Start)
%F a(n) = 5*a(n-) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F E.g.f.: (858 + 2281*x + 3861*x^2 + 990*x^3 + x^4)*exp(x). (End)
%t Table[n^4+984n^3+902n^2+394n+858,{n,0,30}] (* _Harvey P. Dale_, Aug 16 2011 *)
%o (Magma) [n^4+984*n^3+902*n^2+394*n+858: n in [0..30]]; // _Vincenzo Librandi_, Aug 17 2011
%o (PARI) a(n) = n^4+984*n^3+902*n^2+394*n+858 \\ _Charles R Greathouse IV_, Aug 17 2011
%Y Cf. A163303.
%K nonn,easy
%O 0,1
%A _Vincenzo Librandi_, Jul 24 2009
%E Corrected and extended by _Harvey P. Dale_, Aug 16 2011
%E Offset changed from 1 to 0 by _Vincenzo Librandi_, Aug 17 2011