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A190914 Expansion of ( 5-9*x^2-2*x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ). 2
5, 0, 6, 3, 18, 10, 57, 42, 178, 165, 566, 616, 1821, 2236, 5914, 7963, 19362, 27982, 63813, 97394, 211458, 336633, 703786, 1157544, 2350597, 3964960, 7872702, 13541691, 26425522, 46147178, 88853297, 156994354, 299165378, 533410837, 1008343310, 1810544592, 3401446413, 6140811708, 11481472994, 20815538227 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The sequence ..., 14, 29, 10, 2, 9, 2, 0, [5], 0, 6, 3, 18, 10, 57, 42, ...

(the number in square brackets at index 0) equals the trace of:

[ 0 0 0 0-1 ]

[ 1 0 0 0 0 ]

[ 0 1 0 0 1 ]^(+n)

[ 0 0 1 0 3 ]

[ 0 0 0 1 0 ]

or

[ 0 0 0 0-1 ]

[ 1 0 0 0 0 ]

[ 0 1 0 0 3 ]^(-n)

[ 0 0 1 0 1 ]

[ 0 0 0 1 0 ]

Its characteristic polynomial is (x^2 +/- x - 1) * (x^3 -/+ x^2 -/+ x - 1); these factors are fibonacci and tribonacci polynomials.  The ratio of negative terms approaches the golden ratio; the ratio of positive terms approaches the tribonacci constant.

Prime numbers p divide a(+p) and a(-p), as the trace of a matrix M^p (mod p) is constant.

Nonprimes c very rarely divide a(+c) and a(-c) simultaneously.  The only known dual pseudoprime in the sequence is 1.

The distribution of residues induces gaps between pseudoprimes having roughly the size of c.  For example, after 1034881 there is a gap of more than one million terms without either variety of pseudoprime.

Pseudoprimes appear limited to squared primes and squarefree numbers with three or more prime factors.  11 and 13 are more common than other factors.

Positive pseudoprimes: c | a(+c)

----------------------------------------------

1

3481. . . . 59^2

17143 . . . 7 31 79

105589. . . 11 29 331

635335. . . 5 283 449

2992191 . . 3 29 163 211

3659569 . . 1913^2

Negative pseudoprimes: c | a(-c)

----------------------------------------------

1

9 . . . . . 3^2

806 . . . . 2 13 31

1419. . . . 3 11 43

6241. . . . 79^2

6721. . . . 11 13 47

12749 . . . 11 19 61

21106 . . . 2 61 173

38714 . . . 2 13 1489

146689. . . 383^2

649621. . . 7 17 53 103

1034881 . . 41 43 587

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,1,0,-1).

FORMULA

a(n) = A061084(n+1) + A001644(n). - R. J. Mathar, Jun 06 2011

MATHEMATICA

LinearRecurrence[{0, 3, 1, 0, -1}, {5, 0, 6, 3, 18}, 40] (* G. C. Greubel, Apr 23 2019 *)

PROG

(PARI) my(x='x+O('x^40)); Vec((5-9*x^2-2*x^3)/((1+x-x^2)*(1-x-x^2-x^3))) \\ G. C. Greubel, Apr 23 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (5-9*x^2 -2*x^3)/((1+x-x^2)*(1-x-x^2-x^3)) )); // G. C. Greubel, Apr 23 2019

(Sage) ((5-9*x^2-2*x^3)/((1+x-x^2)*(1-x-x^2-x^3))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

CROSSREFS

Cf. A190913 (extended to negative indices), A000045, A000073, A001608, A000040, A005117, A125666.

Sequence in context: A035550 A197508 A159751 * A153458 A096287 A240243

Adjacent sequences:  A190911 A190912 A190913 * A190915 A190916 A190917

KEYWORD

nonn

AUTHOR

Reikku Kulon, May 23 2011

STATUS

approved

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Last modified September 24 08:31 EDT 2021. Contains 347623 sequences. (Running on oeis4.)