Somos's quadratic recurrence constant is defined via the sequence
(1)
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with . This has closed-form solution
(2)
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where is a polylogarithm, is a Lerch transcendent. The first few terms are 1, 2, 12, 576, 1658880, 16511297126400, ... (OEIS A052129). The terms of this sequence have asymptotic growth as
(3)
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(OEIS A116603; Finch 2003, p. 446, term corrected), where is known as Somos's quadratic recurrence constant. Here, the generating function in satisfies the functional equation
(4)
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Expressions for include
(5)
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(6)
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(7)
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(8)
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(9)
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(OEIS A112302; Ramanujan 2000, p. 348; Finch 2003, p. 446; Guillera and Sondow 2005).
Expressions for include
(10)
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(11)
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(12)
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(13)
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(14)
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(OEIS A114124; Finch 2003, p. 446; Guillera and Sondow 2005; J. Borwein, pers. comm., Feb. 6, 2005), where is a polylogarithm.
is also given by the unit square integral
(15)
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(16)
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(Guillera and Sondow 2005).
Ramanujan (1911; 2000, p. 323) proposed finding the nested radical expression
(17)
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which converges to 3. Vijayaraghavan (in Ramanujan 2000, p. 348) gives the justification of his process both in general, and in the particular example of .