Closed forms are known for the sums of reciprocals of even-indexed Lucas numbers
(1)
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(2)
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(3)
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(4)
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(5)
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(OEIS A153415), where is the golden ratio, is a q-polygamma function, and is a Jacobi theta function, and odd-indexed Lucas numbers
(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(OEIS A153416), where is a Lambert series (Borwein and Borwein 1987, pp. 91-92). This gives the reciprocal Lucas constant as
(12)
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(13)
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(14)
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(15)
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(16)
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(OEIS A093540), where is the golden ratio and is a Fibonacci number.
Borwein and Borwein (1987, pp. 94-101) give a number of related beautiful formulas.