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A220838 Tropical version of Somos-4 sequence A006720.

%I A220838 #29 Sep 17 2023 01:36:57
%S A220838 -1,0,0,0,1,1,2,3,3,5,6,7,9,10,12,14,15,18,20,22,25,27,30,33,35,39,42,
%T A220838 45,49,52,56,60,63,68,72,76,81,85,90,95,99,105,110,115,121,126,132,
%U A220838 138,143,150,156,162,169,175,182,189,195,203,210,217,225,232
%N A220838 Tropical version of Somos-4 sequence A006720.
%C A220838 Given the generalized Somos-4 sequence with variables s(1), s(2), s(3), s(4), u, v and recursion s(n) = (u*s(n-1)*s(n-3) + v*s(n-2)^2)/s(n-4), then s(n) is a Laurent polynomial with denominator b(n) := s(1)^a(n)*s(2)^a(n-1)*s(3)^a(n-2)*s(4)^a(n-3) for all n in Z. Moreover, s(n)*b(n) is an irreducible polynomial for all n in Z. - _Michael Somos_, Sep 16 2023
%H A220838 G. C. Greubel, <a href="/A220838/b220838.txt">Table of n, a(n) for n = 1..5000</a>
%H A220838 A. Fordy and A. Hone, <a href="http://arxiv.org/abs/1207.6072">Discrete integrable systems and Poisson algebras from cluster maps</a>, arXiv:1207.6072 [nlin.SI], 2012, See Example 3.6.
%H A220838 A. P. Fordy, <a href="http://arxiv.org/abs/1403.8061">Periodic Cluster Mutations and Related Integrable Maps</a>, arXiv preprint arXiv:1403.8061 [math-ph], 2014.
%H A220838 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,1,-2,1).
%F A220838 From _Michael Somos_, Dec 27 2012: (Start)
%F A220838 G.f.: x * (x^6 - x^5 + x^4 - x^2 + 2*x - 1) / ( (1 - x)^2 * (1 - x^8) ).
%F A220838 a(2-n) = a(n). (End)
%F A220838 Second difference has period 8. - _Michael Somos_, Dec 27 2012
%F A220838 a(n) = A236294(n-5) = max( a(n-1) + a(n-3), 2*a(n-2) ) - a(n-4) for all n in Z. - _Michael Somos_, Sep 16 2023
%e A220838 G.f. = -x + x^5 + x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 5*x^10 + 6*x^11 + 7*x^12 + ...
%p A220838 A118825x := proc(n)
%p A220838     coeftayl((1-2*x+x^2)/(x^4+1),x=0,n) ;
%p A220838 end proc:
%p A220838 A056594 := proc(n)
%p A220838     coeftayl(1/(x^2+1),x=0,n) ;
%p A220838 end proc:
%p A220838 A220838 := proc(n)
%p A220838     -9/32-1/8*n+1/16*n^2+1/32*(-1)^n ;
%p A220838     %+A118825x(n)/4 - A056594(n+3)/8  ;
%p A220838 end proc:
%p A220838 seq(A220838(n),n=0..80) ; # _R. J. Mathar_, Jan 30 2013
%t A220838 LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 1, -2, 1}, {-1, 0, 0, 0, 1, 1, 2, 3, 3, 5}, 62] (* _Jean-François Alcover_, Nov 26 2017 *)
%t A220838 a[ n_] := With[{m = n-1}, Floor[m^2/16] - Boole[Mod[m, 8] == 0]]; (* _Michael Somos_, Sep 16 2023 *)
%o A220838 (PARI) {a(n) = if( n<1, n = 2-n); polcoeff( x * (x^6 - x^5 + x^4 - x^2 + 2*x - 1) / ( (1 - x)^2 * (1 - x^8) ) + x * O(x^n), n)} /* _Michael Somos_, Dec 27 2012 */
%o A220838 (PARI) {a(n) = n--; n^2\16 - !(n%8)}; /* _Michael Somos_, Sep 16 2023 */
%o A220838 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*( x^6-x^5+x^4-x^2+2*x-1)/((1-x)^2*(1-x^8)))); // _G. C. Greubel_, Aug 10 2018
%Y A220838 Cf. A006720, A236294.
%K A220838 sign,easy
%O A220838 1,7
%A A220838 _N. J. A. Sloane_, Dec 23 2012