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A100321
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The trinomial transform (A027907) gives powers of 2, while the trinomial transform of this sequence shift one place left gives powers of 3.
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2
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1, 1, 0, 2, -3, 8, -16, 35, -72, 150, -307, 628, -1276, 2587, -5228, 10546, -21235, 42704, -85784, 172179, -345344, 692286, -1387155, 2778492, -5563748, 11138443, -22294596, 44617850, -89282067, 178639160, -357399712, 714995843, -1430309496, 2861133222, -5723098483, 11447543236
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: (1 + 3*x - 3*x^3) / (1 + 2*x - 2*x^2 - 3*x^3 + 2*x^4).
2^n = Sum_{k=0..2*n} A027907(n, k)*a(k).
3^n = Sum_{k=0..2*n} A027907(n, k)*a(k+1).
a(n) = (1/3)*((-1)^n*(3*Fibonacci(n-1) - 2^n) + 1). - Ralf Stephan, May 15 2007
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EXAMPLE
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2^3 = 1*(1) + 3*(1) + 6*(0) + 7*(2) + 6*(-3) + 3*(8) + 1*(-16).
3^3 = 1*(1) + 3*(0) + 6*(2) + 7*(-3) + 6*(8) + 3*(-16) + 1*(35).
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MATHEMATICA
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LinearRecurrence[{-2, 2, 3, -2}, {1, 1, 0, 2}, 41] (* G. C. Greubel, Feb 01 2023 *)
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PROG
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(PARI) a(n)=polcoeff((1+3*x-3*x^3)/(1+2*x-2*x^2-3*x^3+2*x^4+x*O(x^n)), n)
(Magma) [((-1)^n*(3*Fibonacci(n-1) -2^n) +1)/3: n in [0..40]]; // G. C. Greubel, Feb 01 2023
(SageMath)
def A100321(n): return ((-1)^n*(3*fibonacci(n-1) -2^n) +1)/3
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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