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A106515
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A Fibonacci-Pell convolution.
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4
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1, 2, 6, 15, 38, 94, 231, 564, 1372, 3329, 8064, 19512, 47177, 114010, 275430, 665247, 1606534, 3879302, 9366735, 22615356, 54601628, 131825377, 318263328, 768369744, 1855031473, 4478479058, 10812064614, 26102729679, 63017720390
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listen;
history;
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internal format)
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1-x)/((1-x-x^2)*(1-2*x-x^2)).
a(n) = Sum_{k=0..n} Fibonacci(n-k-1)*Pell(k+1).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor((n-k+1)/2)} binomial(n-k+1, 2*j+k+1)*2^j.
a(n) = Pell(n) + Pell(n+1) - Fibonacci(n). - Ralf Stephan, Jun 02 2007
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MATHEMATICA
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Table[Fibonacci[n, 2] + Fibonacci[n+1, 2] - Fibonacci[n], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)
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PROG
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(Magma)
Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
[Pell(n) + Pell(n+1) - Fibonacci(n): n in [0..30]]; // G. C. Greubel, Aug 05 2021
(Sage) [lucas_number1(n+1, 2, -1) + lucas_number1(n, 2, -1) - lucas_number1(n, 1, -1) for n in (0..30)] # G. C. Greubel, Aug 05 2021
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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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