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A133586
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Expansion of x*(1+2*x)/( (x^2-x-1)*(x^2+x-1) ).
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2
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1, 2, 3, 6, 8, 16, 21, 42, 55, 110, 144, 288, 377, 754, 987, 1974, 2584, 5168, 6765, 13530, 17711, 35422, 46368, 92736, 121393, 242786, 317811, 635622, 832040, 1664080, 2178309, 4356618, 5702887, 11405774, 14930352, 29860704, 39088169, 78176338, 102334155
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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For odd-indexed terms, a(n) = F(n+1). For even-indexed terms, a(n) = 2*a(n-1).
a(n) = (2^(-2-n)*((1-sqrt(5))^n*(-5+sqrt(5)) - (-1-sqrt(5))^n*(-3+sqrt(5)) - (-1+sqrt(5))^n*(3+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5))))/sqrt(5). - Colin Barker, Mar 28 2016
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EXAMPLE
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a(5) = F(6) = 8.
a(6) = 2*a(5) = 2*8 = 16.
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MAPLE
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A133586aux := proc(n, k)
end proc:
combinat[fibonacci](n) ;
end proc:
add(A133586aux(n, j)*A000045(j), j=0..n) ;
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MATHEMATICA
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CoefficientList[Series[(1 + 2 x)/((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 21 2015 *)
LinearRecurrence[{0, 3, 0, -1}, {1, 2, 3, 6}, 40] (* Harvey P. Dale, Dec 10 2017 *)
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PROG
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(PARI) {a(n) = if( n%2, fibonacci(n+1), 2*fibonacci(n))}; /* Michael Somos, Jun 20 2015 */
(PARI) Vec(x*(1+2*x)/((x^2-x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Mar 28 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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New definition and A-number in previous definition corrected by R. J. Mathar, Jun 20 2015
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STATUS
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approved
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