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A163303
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a(n) = n^3 + 73*n^2 + n + 67.
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2
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67, 142, 369, 754, 1303, 2022, 2917, 3994, 5259, 6718, 8377, 10242, 12319, 14614, 17133, 19882, 22867, 26094, 29569, 33298, 37287, 41542, 46069, 50874, 55963, 61342, 67017, 72994, 79279, 85878, 92797, 100042, 107619, 115534, 123793, 132402
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OFFSET
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0,1
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COMMENTS
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Sequences generated by primitive polynomial J(p)=J(1031), for k=3.
Comment (entirely taken from Cugiani's text - see References) from Vincenzo Librandi , Aug 23 2011: (Start)
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)
i) (-1)^k a(0) is a primitive element of J(p).
ii) The remainder of the division of x^r through the polynomial equals (-1)^k a(0).
iii) The remainder of the division of x^(r/q) through the polynomial must have positive degree for each prime divisor q|r.
(End)
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REFERENCES
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Marco Cugiani, Metodi numerico statistici (Collezione di Matematica applicata n.7), UTET Torino, 1980, pp. 78-84
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LINKS
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FORMULA
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G.f.: ( 67-126*x+203*x^2-138*x^3 ) / (x-1)^4 . - R. J. Mathar, Aug 21 2011
E.g.f: (67 + 75*x + 76*x^2 + x^3)*exp(x). - G. C. Greubel, Dec 18 2016
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MATHEMATICA
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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 *)
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PROG
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(Magma) [n^3+73*n^2+n+67: n in [0..40]];
(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
(PARI) first(m)=vector(m, i, i--; i^3 + 73*i^2 + i + 67) \\ Anders Hellström, Sep 13 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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