A086313 - OEIS

%I #14 Jun 27 2014 16:17:16

%S 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,

%T 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,

%U 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

%N Decimal expansion of constant c appearing in the expected number of pair of twin vacancies in a digital tree.

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TreeSearching.html">Tree Searching</a>

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

%e 0.37204...

%t 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 *)

%Y Cf. A048651, A065442, A086315.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Jul 15 2003