login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A086315 Decimal expansion of constant theta appearing in the expected number of pair of twin vacancies in a digital tree. 1
7, 7, 4, 3, 1, 3, 1, 9, 8, 5, 5, 3, 6, 8, 9, 6, 5, 9, 1, 4, 4, 6, 2, 8, 3, 8, 5, 6, 7, 4, 9, 7, 8, 4, 2, 9, 5, 5, 9, 3, 6, 5, 2, 8, 2, 8, 4, 1, 8, 8, 0, 8, 8, 8, 8, 6, 6, 5, 1, 8, 5, 5, 9, 1, 8, 3, 8, 3, 2, 9, 9, 7, 1, 5, 1, 7, 6, 2, 9, 2, 9, 0, 1, 5, 1, 0, 9, 4, 3, 9, 0, 7, 9, 9, 5, 5, 4, 3, 5, 6, 7, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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=1..102.

Eric Weisstein's World of Mathematics, Tree Searching

FORMULA

Sum_{k>=1} k*2^(k*(k-1)/2))*(Sum_{j=1..k} 1/(2^j-1))/Product_{j=1..k} (2^j-1).

EXAMPLE

7.743131985536896591446283856749784...

MAPLE

theta:= sum((((k*2^(k*(k-1)/2)) *sum(1/(2^j-1), j=1..k))/

        product(2^j-1, j=1..k)), k=1..infinity):

s:= convert(evalf(theta, 110), string):

map(parse, subs("."=[][], [seq(i, i=s)]))[]; # Alois P. Heinz, Jun 27 2014

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], Print["m = ", m]; m = m + dm]; RealDigits[theta[m], 10, digits] // First (* Jean-François Alcover, Jun 27 2014 *)

digits = 102; theta = NSum[((k*2^(k*((k-1)/2)))*((QPolyGamma[0, 1+k, 1/2] - QPolyGamma[0, 1, 1/2])/Log[2]))/((-1)^k*QPochhammer[2, 2, k]), {k, 1, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 3*digits]; RealDigits[theta, 10, digits] // First (* Jean-François Alcover, Nov 19 2015 *)

CROSSREFS

Sequence in context: A265306 A198544 A247274 * A185577 A201517 A010513

Adjacent sequences:  A086312 A086313 A086314 * A086316 A086317 A086318

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Jul 15 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 5 18:00 EDT 2021. Contains 346488 sequences. (Running on oeis4.)