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A091870
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A trinomial transform of Fibonacci(3n).
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4
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0, 1, 8, 53, 336, 2105, 13144, 81997, 511392, 3189169, 19888040, 124023461, 773419248, 4823095913, 30077155576, 187563189565, 1169656805184, 7294059356257, 45486249993032, 283655347025429, 1768894026280080
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OFFSET
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0,3
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COMMENTS
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Binomial transform of A084326.
Second binomial transform of A001076(n) = Fibonacci(3n)/2.
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..1258
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (8,-11).
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FORMULA
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G.f.: x/(1 - 8*x + 11*x^2).
a(n) = sqrt(5) * ((4+sqrt(5))^n - (4-sqrt(5))^n) / 10.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (n!/(i!*j!*(n-i-j)!)) * Fibonacci(3*i) / 2.
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MATHEMATICA
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LinearRecurrence[{8, -11}, {0, 1}, 30] (* G. C. Greubel, May 21 2019 *)
CoefficientList[Series[x/(1 - 8 x + 11 x^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 22 2017 *)
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PROG
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(Sage) [lucas_number1(n, 8, 11) for n in xrange(0, 30)] # Zerinvary Lajos, Apr 23 2009
(PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1 -8*x +11*x^2))) \\ G. C. Greubel, May 21 2019
(MAGMA) [n le 2 select n-1 else 8*Self(n-1) -11*Self(n-2): n in [1..30]]; // G. C. Greubel, May 21 2019
(GAP) a:=[0, 1];; for n in [3..30] do a[n]:=8*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, May 21 2019
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CROSSREFS
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Cf. A084326.
Sequence in context: A291662 A110099 A297334 * A252824 A054418 A293115
Adjacent sequences: A091867 A091868 A091869 * A091871 A091872 A091873
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KEYWORD
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nonn,easy
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AUTHOR
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Paul Barry, Feb 06 2004
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STATUS
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approved
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