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A333251
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Tropical version of Somos-5 sequence A006721.
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2
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-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, 27, 30, 32, 34, 36, 39, 41, 43, 46, 49, 51, 54, 57, 60, 62, 66, 69, 72, 75, 79, 82, 85, 89, 93, 96, 100, 104, 108, 111, 116, 120, 124, 128, 133, 137, 141, 146, 151
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OFFSET
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0,9
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COMMENTS
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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).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,0,1,-1,-1,1).
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FORMULA
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G.f.: -(1 - x - x^2 + x^3 - x^5)/((1 - x)*(1 - x^2)*(1 - x^7)).
a(n) = max( a(n-1) + a(n-4), a(n-2) + a(n-3) ) - a(n-5) for all n in Z.
a(n) = a(n+7) - 2 - floor(n/2) for all n in Z.
Second difference has period 14.
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EXAMPLE
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G.f. = -1 + x^5 + x^6 + x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 4*x^11 + ...
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).
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MATHEMATICA
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a[ n_] := With[{m = Mod[n, 14]}, Quotient[n^2, 28] - Boole[m==0] + Boole[m==5] + Boole[m==9]];
a[ n_] := SeriesCoefficient[ -(1 - x - x^2 + x^3 - x^5) / ((1 - x) * (1 - x^2) * (1 - x^7)), {x, 0, Abs@n}];
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PROG
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(PARI) {a(n) = n^2\28 - (n%14==0) + (n%14==5) + (n%14==9)};
(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)};
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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