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Expansion of ( 5-9*x^2-2*x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).
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%I #22 May 06 2024 06:54:18

%S 5,0,6,3,18,10,57,42,178,165,566,616,1821,2236,5914,7963,19362,27982,

%T 63813,97394,211458,336633,703786,1157544,2350597,3964960,7872702,

%U 13541691,26425522,46147178,88853297,156994354,299165378,533410837,1008343310,1810544592,3401446413,6140811708,11481472994,20815538227

%N Expansion of ( 5-9*x^2-2*x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).

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

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

%C [ 0 0 0 0-1 ]

%C [ 1 0 0 0 0 ]

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

%C [ 0 0 1 0 3 ]

%C [ 0 0 0 1 0 ]

%C or

%C [ 0 0 0 0-1 ]

%C [ 1 0 0 0 0 ]

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

%C [ 0 0 1 0 1 ]

%C [ 0 0 0 1 0 ]

%C 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.

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

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

%C 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.

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

%C Positive pseudoprimes: c | a(+c)

%C ----------------------------------------------

%C 1

%C 3481. . . . 59^2

%C 17143 . . . 7 31 79

%C 105589. . . 11 29 331

%C 635335. . . 5 283 449

%C 2992191 . . 3 29 163 211

%C 3659569 . . 1913^2

%C Negative pseudoprimes: c | a(-c)

%C ----------------------------------------------

%C 1

%C 9 . . . . . 3^2

%C 806 . . . . 2 13 31

%C 1419. . . . 3 11 43

%C 6241. . . . 79^2

%C 6721. . . . 11 13 47

%C 12749 . . . 11 19 61

%C 21106 . . . 2 61 173

%C 38714 . . . 2 13 1489

%C 146689. . . 383^2

%C 649621. . . 7 17 53 103

%C 1034881 . . 41 43 587

%H G. C. Greubel, <a href="/A190914/b190914.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1,0,-1).

%F a(n) = A061084(n+1) + A001644(n). - _R. J. Mathar_, Jun 06 2011

%t LinearRecurrence[{0, 3, 1, 0, -1}, {5, 0, 6, 3, 18}, 40] (* _G. C. Greubel_, Apr 23 2019 *)

%o (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

%o (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

%o (SageMath) ((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

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

%K nonn,easy

%O 0,1

%A _Reikku Kulon_, May 23 2011