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

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

Offset: 0

Author

Vincenzo Librandi, Jul 24 2009, Jul 25 2009

Keywords

• nonn
• easy

Comments

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)

References

• Marco Cugiani, Metodi numerico statistici (Collezione di Matematica applicata n.7), UTET Torino, 1980, pp. 78-84

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Tom Hansen, G. L. Mullen, Primitive Polynomials over finite fields, Math. Comp. 59 (200) (1992) 639
• Sean E. O'Connor, Computing primitive Polynomials - Theory and Algorithm
• Eric Weisstein, MathWorld: Primitive Polynomial
• Wikipedia, Primitive Polynomial
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A163304.

Programs

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

Mathematica

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 *)

PARI

first(m)=vector(m,i,i--;i^3 + 73*i^2 + i + 67) \\ Anders Hellström, Sep 13 2015

Formula

G.f.: ( 67-126*x+203*x^2-138*x^3 ) / (x-1)^4 . - R. J. Mathar, Aug 21 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Sep 13 2015

E.g.f: (67 + 75*x + 76*x^2 + x^3)*exp(x). - G. C. Greubel, Dec 18 2016