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A319200
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a(n) = -(A(n) - A(n-1)) where A(n) = A057597(n+1), for n >= 0.
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3
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0, -1, 2, -1, -2, 5, -4, -3, 12, -13, -2, 27, -38, 9, 56, -103, 56, 103, -262, 215, 150, -627, 692, 85, -1404, 2011, -522, -2893, 5426, -3055, -5264, 13745, -11536, -7473, 32754, -36817, -3410, 72981, -106388, 29997, 149372, -285757, 166382, 268747, -720886, 618521, 371112, -1710519, 1957928, 123703, -3792150
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OFFSET
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0,3
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COMMENTS
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This sequence appears in the reduction formula for negative powers of the tribonacci constant t = A058265: t^(-n) = A(n)*t^2 + a(n)*t + A(n+1)*1, with A(n) = A057597(n+1), for n >= 0. This follows from t^3 = t^2 + t + 1, or 1/t = t^2 - t - 1 = A192918, leading to the recurrence: A(n) = -A(n) - A(n-1) + A(n-2), with inputs A(-3) = 1, A(-2) = 1 and A(-1) = 0 and a(n) = -(A(n) - A(n-1)). See the example below.
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LINKS
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FORMULA
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Recurrence a(n) = -a(n-1) - a(n-2) + a(n-3), for n >=0, with a(-3) = 1, a(-2) = 0 and a(-1) = 1.
G.f.: (1 + 1/x)/(1 + x + x^2 - x^3).
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EXAMPLE
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The coefficients of t^2, t, 1 for t^(-n) begin, for n >= -3:
n t^2 t 1
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-3 1 1 1
-2 1 0 0
-1 0 1 0
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+0 0 0 1
+1 1 -1 -1
+2 -1 2 0
+3 0 -1 2
+4 2 -2 -3
+5 -3 5 1
+6 1 -4 4
+7 4 -3 -8
+8 -8 12 5
+9 5 -13 7
10 7 -2 -20
...
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PROG
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(PARI) a057597(n) = polcoeff( if( n<0, x / ( 1 - x - x^2 - x^3), x^2 / ( 1 + x + x^2 - x^3) ) + x*O(x^abs(n)), abs(n)) \\ after Michael Somos in A057597
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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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