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A084170
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a(n) = (5*2^n + (-1)^n - 3)/3.
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6
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1, 2, 6, 12, 26, 52, 106, 212, 426, 852, 1706, 3412, 6826, 13652, 27306, 54612, 109226, 218452, 436906, 873812, 1747626, 3495252, 6990506, 13981012, 27962026, 55924052, 111848106, 223696212, 447392426, 894784852, 1789569706, 3579139412
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OFFSET
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0,2
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COMMENTS
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Original name of this sequence: Generalized Jacobsthal numbers.
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
a(n) = a(n-1) + 2*a(n-2) + 2, a(0)=1, a(1)=2.
G.f.: (1+x^2)/((1+x)*(1-x)*(1-2*x)).
E.g.f.: 5*exp(2*x)/3 - exp(x) + exp(-x)/3.
a(n) = A169969(2n) - 1, n >= 1; a(n) = 3*2^(n-1) - 1 + A169969(2n-7), n >= 5.
a(n+3) = 15*2^n - 2 - a(n), n >= 0, a(0)=1, a(1)=2, a(2)=6.
a(n) + A026644(n) = 3*2^n - 2, n >= 1.
a(n+3) = 3*2^(n+2) + A026644(n), n >= 1. (End)
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MATHEMATICA
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LinearRecurrence[{2, 1, -2}, {1, 2, 6}, 40] (* or *) Table[(5*2^n+(-1)^n-3)/3, {n, 0, 40}] (* Harvey P. Dale, Jan 29 2012 *)
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PROG
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(SageMath) [(2/3)*(5*2^(n-1) -1 -(n%2)) for n in range(41)] # G. C. Greubel, Oct 11 2022
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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