A163304 -id:A163304 - cp's OEIS frontend

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

Vincenzo Librandi, Jul 24 2009, Jul 25 2009

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)

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

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).

Cf. A163304.

Magma
```
[n^3+73*n^2+n+67: n in [0..40]];
```

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

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 *)

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula