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A167193
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a(n) = (1/3)*(1 - (-2)^n + 3*(-1)^n ) = (-1)^(n+1)*A167030(n).
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1
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1, 0, 0, 2, -4, 10, -20, 42, -84, 170, -340, 682, -1364, 2730, -5460, 10922, -21844, 43690, -87380, 174762, -349524, 699050, -1398100, 2796202, -5592404, 11184810, -22369620, 44739242, -89478484, 178956970, -357913940, 715827882, -1431655764, 2863311530, -5726623060, 11453246122, -22906492244, 45812984490
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OFFSET
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0,4
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COMMENTS
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This is the inverse binomial transform of 1, 1, 1, 3, 5, 11,.. (continued as in A001045 and conjectured to be equal to A152046).
Any sequence (like this one) which obeys a(n)= -2a(n-1)+a(n-2)+2a(n-3) also obeys a(n)=5a(n-2)-4a(n-4), proved by telescoping; see A101622.
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LINKS
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FORMULA
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G.f.: ( -1 - 2*x + x^2 ) / ( (x-1)*(1+2*x)*(1+x) ).
a(n) = -a(n-1) + 2*a(n-2) - 2*(-1)^n.
a(n) = -2*a(n-1) + a(n-2) + 2*a(n-3).
E.g.f.: (1/3)*(exp(x) + 3*exp(-x) - exp(-2*x)). - G. C. Greubel, Jun 04 2016
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MATHEMATICA
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LinearRecurrence[{-2, 1, 2}, {1, 0, 0}, 25] (* or *) Table[(1/3)*(1 + 3*(-1)^n - (-2)^n), {n, 0, 25}] (* G. C. Greubel, Jun 04 2016 *)
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PROG
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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