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Search: id:A166168
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| A166168 |
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G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n). |
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+0
3
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| 1, 1, 4, 29, 585, 34212, 5600397, 2490542953, 2968152042068, 9416588994339205, 79216509536543420965, 1762508872870620792746360, 103525263562786817866762466405, 16031370626878431551103688398524485
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Conjectured to consist entirely of integers.
The Lucas numbers (A000204) forms the logarithmic derivative of the Fibonacci numbers (A000045).
Note that Lucas(n^2) = [(1+sqrt(5))/2]^(n^2) + [(1-sqrt(5))/2]^(n^2).
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FORMULA
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a(n) = (1/n)*Sum_{k=1..n} Lucas(k^2)*a(n-k), a(0)=1.
Logarithmic derivative yields A166169.
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 585*x^4 + 34212*x^5 +...
log(A(x)) = x + 7*x^2/2 + 76*x^3/3 + 2207*x^4/4 + 167761*x^5/5 + 33385282*x^6/6 +...+ Lucas(n^2)*x^n/n +...
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PROGRAM
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(PARI) {a(n)=polcoeff(exp(sum(m=1, n, (fibonacci(m^2-1)+fibonacci(m^2+1))*x^m/m)+x*O(x^n)), n)}
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CROSSREFS
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Cf. A166169, A156216, A155200, A000204, A000045.
Sequence in context: A000798 A135485 A162287 this_sequence A126559 A159862 A164820
Adjacent sequences: A166165 A166166 A166167 this_sequence A166169 A166170 A166171
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Oct 08 2009
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