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A086313 Decimal expansion of constant c appearing in the expected number of pair of twin vacancies in a digital tree. 0
3, 7, 2, 0, 4, 8, 6, 8, 1, 2, 0, 7, 7, 7, 1, 8, 3, 3, 2, 4, 2, 0, 5, 4, 2, 5, 4, 0, 3, 0, 1, 2, 8, 8, 9, 2, 2, 7, 6, 1, 5, 2, 9, 5, 8, 9, 4, 0, 7, 6, 2, 9, 7, 8, 7, 9, 4, 3, 3, 1, 2, 2, 8, 5, 2, 2, 3, 7, 3, 2, 3, 8, 0, 8, 5, 6, 9, 7, 3, 0, 5, 5, 0, 7, 0, 3, 0, 6, 6, 0, 8, 0, 3, 2, 8, 8, 0, 9, 8, 9, 0, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.13 Binary search tree constants, p. 356.

LINKS

Table of n, a(n) for n=0..101.

Eric Weisstein's World of Mathematics, Tree Searching

FORMULA

c = theta + 1 - 1/Q*(1/log(2) + alpha^2 - alpha), where theta is A086315, Q is A048651 and alpha is A065442.

EXAMPLE

0.37204...

MATHEMATICA

digits = 102; m0 = 100; dm = 100; Clear[theta]; theta[m_] := theta[m] = Sum[((k*2^(k*((k-1)/2)))*Sum[1/(2^j-1), {j, 1, k}])/Product[2^j-1, {j, 1, k}], {k, 1, m}] // N[#, digits+10]&; theta[m0]; theta[m = m0+dm]; While[RealDigits[theta[m], 10, digits+10] != RealDigits[theta[m-dm], 10, digits+10], m = m+dm]; theta0 = theta[m]; Clear[Q]; Q[m_] := Q[m] = Product[1-1/2^k, {k, 1, m}] // N[#, digits+10]&; Q[m0]; Q[m = m0+dm]; While[RealDigits[Q[m], 10, digits+10] != RealDigits[Q[m-dm], 10, digits+10], m = m+dm]; Q0 = Q[m]; Clear[alpha]; alpha[m_] := alpha[m] = Sum[1/(2^k-1), {k, 1, m}] // N[#, digits+10]&; alpha[m0]; alpha[m = m0+dm]; While[RealDigits[alpha[m], 10, digits+10] != RealDigits[alpha[m-dm], 10, digits+10], m = m+dm]; alpha0 = alpha[m]; c = theta0+1-1/Q0*(1/Log[2]+alpha0^2-alpha0); RealDigits[c, 10, digits] // First (* Jean-François Alcover, Jun 27 2014 *)

CROSSREFS

Cf. A048651, A065442, A086315.

Sequence in context: A144556 A001439 A318215 * A016451 A093097 A086465

Adjacent sequences:  A086310 A086311 A086312 * A086314 A086315 A086316

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Jul 15 2003

STATUS

approved

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Last modified November 17 05:27 EST 2019. Contains 329217 sequences. (Running on oeis4.)