cp's OEIS Frontend

A163303 a(n) = n^3 + 73*n^2 + n + 67.

%I A163303 #50 Feb 16 2025 08:33:11
%S A163303 67,142,369,754,1303,2022,2917,3994,5259,6718,8377,10242,12319,14614,
%T A163303 17133,19882,22867,26094,29569,33298,37287,41542,46069,50874,55963,
%U A163303 61342,67017,72994,79279,85878,92797,100042,107619,115534,123793,132402
%N A163303 a(n) = n^3 + 73*n^2 + n + 67.
%C A163303 Sequences generated by primitive polynomial J(p)=J(1031), for k=3.
%C A163303 Comment (entirely taken from Cugiani's text - see References) from _Vincenzo Librandi_ , Aug 23 2011: (Start)
%C A163303 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 A163303   i) (-1)^k a(0) is a primitive element of J(p).
%C A163303   ii) The remainder of the division of x^r through the polynomial equals (-1)^k a(0).
%C A163303   iii) The remainder of the division of x^(r/q) through the polynomial must have positive degree for each prime divisor q|r.
%C A163303 (End)
%D A163303 Marco Cugiani, Metodi numerico statistici (Collezione di Matematica applicata n.7), UTET Torino, 1980, pp. 78-84
%H A163303 Vincenzo Librandi, <a href="/A163303/b163303.txt">Table of n, a(n) for n = 0..10000</a>
%H A163303 Tom Hansen, G. L. Mullen, <a href="http://www.jstor.org/stable/2153081">Primitive Polynomials over finite fields</a>, Math. Comp. 59 (200) (1992) 639
%H A163303 Sean E. O'Connor, <a href="http://seanerikoconnor.freeservers.com/Mathematics/AbstractAlgebra/PrimitivePolynomials/theory.html">Computing primitive Polynomials - Theory and Algorithm</a>
%H A163303 Eric Weisstein, <a href="https://mathworld.wolfram.com/PrimitivePolynomial.html">MathWorld: Primitive Polynomial</a>
%H A163303 Wikipedia, <a href="http://en.wikipedia.org/wiki/Primitive_polynomial">Primitive Polynomial</a>
%H A163303 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A163303 G.f.: ( 67-126*x+203*x^2-138*x^3 ) / (x-1)^4 . - _R. J. Mathar_, Aug 21 2011
%F A163303 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Sep 13 2015
%F A163303 E.g.f: (67 + 75*x + 76*x^2 + x^3)*exp(x). - _G. C. Greubel_, Dec 18 2016
%t A163303 Table[n^3 + 73 n^2 + n + 67, {n, 0, 60}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {67, 142, 369, 754}, 50] (* _Vincenzo Librandi_, Sep 13 2015 *)
%o A163303 (Magma) [n^3+73*n^2+n+67: n in [0..40]];
%o A163303 (Magma) I:=[67,142,369,754]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..60]]; // _Vincenzo Librandi_, Sep 13 2015
%o A163303 (PARI) first(m)=vector(m,i,i--;i^3 + 73*i^2 + i + 67) \\ _Anders Hellström_, Sep 13 2015
%Y A163303 Cf. A163304.
%K A163303 nonn,easy
%O A163303 0,1
%A A163303 _Vincenzo Librandi_, Jul 24 2009, Jul 25 2009