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A333251 Tropical version of Somos-5 sequence A006721.

%I A333251 #12 Sep 24 2021 11:31:42
%S A333251 -1,0,0,0,0,1,1,1,2,3,3,4,5,6,6,8,9,10,11,13,14,15,17,19,20,22,24,26,
%T A333251 27,30,32,34,36,39,41,43,46,49,51,54,57,60,62,66,69,72,75,79,82,85,89,
%U A333251 93,96,100,104,108,111,116,120,124,128,133,137,141,146,151
%N A333251 Tropical version of Somos-5 sequence A006721.
%C A333251 If (x, y, s(0), .., s(4)) are 7 variables and s(n) = (x*s(n-1)*s(n-4) + y*s(n-2)*s(n-3))/s(n-5) for n>=5 is the generalized Somos-5 sequence, then s(n) is a Laurent polynomial in the variables with the numerator being irreducible and the denominator is Product_{k=0..4} s(k)^a(n-k).
%H A333251 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,0,1,-1,-1,1).
%F A333251 G.f.: -(1 - x - x^2 + x^3 - x^5)/((1 - x)*(1 - x^2)*(1 - x^7)).
%F A333251 a(n) = max( a(n-1) + a(n-4), a(n-2) + a(n-3) ) - a(n-5) for all n in Z.
%F A333251 a(n) = a(n+7) - 2 - floor(n/2) for all n in Z.
%F A333251 Second difference has period 14.
%e A333251 G.f. = -1 + x^5 + x^6 + x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 4*x^11 + ...
%e A333251 s(7) = ((s(0)*s(3)^2*s(4) + s(1)^2*s(4)^2)*x*y + s(1)*s(2)*s(3)*s(4)*(y^2+x^3) + s(2)^2*s(3)^2*x^2*y)/(s(0)^1*s(1)^1*s(2)^1*s(3)^0*s(4)^0).
%t A333251 a[ n_] := With[{m = Mod[n, 14]}, Quotient[n^2, 28] - Boole[m==0] + Boole[m==5] + Boole[m==9]];
%t A333251 a[ n_] := SeriesCoefficient[ -(1 - x - x^2 + x^3 - x^5) / ((1 - x) * (1 - x^2) * (1 - x^7)), {x, 0, Abs@n}];
%o A333251 (PARI) {a(n) = n^2\28 - (n%14==0) + (n%14==5) + (n%14==9)};
%o A333251 (PARI) {a(n) = n=abs(n); polcoeff( -(1 - x - x^2 + x^3 - x^5) / ((1 - x) * (1 - x^2) * (1 - x^7)) + x * O(x^n), n)};
%Y A333251 Cf. A006721, A220838.
%K A333251 sign,easy
%O A333251 0,9
%A A333251 _Michael Somos_, Mar 13 2020