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A014162
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Apply partial sum operator thrice to Fibonacci numbers.
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13
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0, 1, 4, 11, 25, 51, 97, 176, 309, 530, 894, 1490, 2462, 4043, 6610, 10773, 17519, 28445, 46135, 74770, 121115, 196116, 317484, 513876, 831660, 1345861, 2177872, 3524111, 5702389, 9226935, 14929789
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OFFSET
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0,3
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COMMENTS
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With offset 4, number of 132-avoiding two-stack sortable permutations which contain exactly one subsequence of type 51234.
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LINKS
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FORMULA
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G.f.: x/((1-x)^3*(1-x-x^2)).
a(n-2) = Sum_{k=0..floor(n/2)} binomial(n-k, k+3).
a(n-2) = Sum_{k=0..n} binomial(k, n-k+3). (End)
a(n) = Fibonacci(n+6) - (n^2 + 7*n + 16)/2.
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MAPLE
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with(combinat); seq(fibonacci(n+6)-(n^2+7*n+16)*(1/2), n = 0..40); # G. C. Greubel, Sep 05 2019
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MATHEMATICA
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Nest[Accumulate, Fibonacci[Range[0, 30]], 3] (* or *) LinearRecurrence[{4, -5, 1, 2, -1}, {0, 1, 4, 11, 25}, 40] (* Harvey P. Dale, Aug 19 2017 *)
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PROG
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(Magma) [Fibonacci(n+6) - (n^2 + 7*n + 16)/2: n in [0..40]]; // G. C. Greubel, Sep 05 2019
(Sage) [fibonacci(n+6) - (n^2 + 7*n + 16)/2 for n in (0..40)] # G. C. Greubel, Sep 05 2019
(GAP) List([0..40], n-> Fibonacci(n+6) - (n^2 + 7*n + 16)/2); # G. C. Greubel, Sep 05 2019
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CROSSREFS
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Right-hand column 6 of triangle A011794.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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