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A190914 Expansion of ( 5-9*x^2-2*x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).

%I A190914 #22 May 06 2024 06:54:18
%S A190914 5,0,6,3,18,10,57,42,178,165,566,616,1821,2236,5914,7963,19362,27982,
%T A190914 63813,97394,211458,336633,703786,1157544,2350597,3964960,7872702,
%U A190914 13541691,26425522,46147178,88853297,156994354,299165378,533410837,1008343310,1810544592,3401446413,6140811708,11481472994,20815538227
%N A190914 Expansion of ( 5-9*x^2-2*x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).
%C A190914 The sequence ..., 14, 29, 10, 2, 9, 2, 0, [5], 0, 6, 3, 18, 10, 57, 42, ...
%C A190914 (the number in square brackets at index 0) equals the trace of:
%C A190914   [ 0 0 0 0-1 ]
%C A190914   [ 1 0 0 0 0 ]
%C A190914   [ 0 1 0 0 1 ]^(+n)
%C A190914   [ 0 0 1 0 3 ]
%C A190914   [ 0 0 0 1 0 ]
%C A190914 or
%C A190914   [ 0 0 0 0-1 ]
%C A190914   [ 1 0 0 0 0 ]
%C A190914   [ 0 1 0 0 3 ]^(-n)
%C A190914   [ 0 0 1 0 1 ]
%C A190914   [ 0 0 0 1 0 ]
%C A190914 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 A190914 Prime numbers p divide a(+p) and a(-p), as the trace of a matrix M^p (mod p) is constant.
%C A190914 Nonprimes c very rarely divide a(+c) and a(-c) simultaneously.  The only known dual pseudoprime in the sequence is 1.
%C A190914 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 A190914 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 A190914 Positive pseudoprimes: c | a(+c)
%C A190914 ----------------------------------------------
%C A190914 1
%C A190914 3481. . . . 59^2
%C A190914 17143 . . . 7 31 79
%C A190914 105589. . . 11 29 331
%C A190914 635335. . . 5 283 449
%C A190914 2992191 . . 3 29 163 211
%C A190914 3659569 . . 1913^2
%C A190914 Negative pseudoprimes: c | a(-c)
%C A190914 ----------------------------------------------
%C A190914 1
%C A190914 9 . . . . . 3^2
%C A190914 806 . . . . 2 13 31
%C A190914 1419. . . . 3 11 43
%C A190914 6241. . . . 79^2
%C A190914 6721. . . . 11 13 47
%C A190914 12749 . . . 11 19 61
%C A190914 21106 . . . 2 61 173
%C A190914 38714 . . . 2 13 1489
%C A190914 146689. . . 383^2
%C A190914 649621. . . 7 17 53 103
%C A190914 1034881 . . 41 43 587
%H A190914 G. C. Greubel, <a href="/A190914/b190914.txt">Table of n, a(n) for n = 0..1000</a>
%H A190914 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1,0,-1).
%F A190914 a(n) = A061084(n+1) + A001644(n). - _R. J. Mathar_, Jun 06 2011
%t A190914 LinearRecurrence[{0, 3, 1, 0, -1}, {5, 0, 6, 3, 18}, 40] (* _G. C. Greubel_, Apr 23 2019 *)
%o A190914 (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 A190914 (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 A190914 (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 A190914 Cf. A190913 (extended to negative indices), A000045, A000073, A001608, A000040, A005117, A125666.
%K A190914 nonn,easy
%O A190914 0,1
%A A190914 _Reikku Kulon_, May 23 2011