Tree Searching

Author

Eric W. Weisstein
August 1, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputerScience/TreeSearching.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TreeSearching.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Constants

α==1-((QPolyGamma3[0, 1, (1/2)])/(log(2)))

V8

α=Sum[(1/((2^k)-1)),{k,∞}]//FullSimplify

1-((QPolyGamma3[0, 1, 1/2])/(log(2)))

N[α]

NSum[(1/((2^k)-1)),{k,∞},WorkingPrecision->100]

β==(((QPolyGamma3[1, 1, (1/2)])+(QPolyGamma3[0, 1, (1/2)]) log(2))/((log^2)(2)))-1

V8

β=Sum[(1/((((2^k)-1))^2)),{k,∞}]//FullSimplify

(((QPolyGamma3[1, 1, 1/2])+(QPolyGamma3[0, 1, 1/2]) log(2))/((log^2)(2)))-1

N[%]

NSum[(1/((((2^k)-1))^2)),{k,∞},WorkingPrecision->100,Method->Fit]

Number of Comparisons

Successful

e1=((EulerGamma-1)/(Log[2]))-α+3/2

Unsuccessful

e2=(EulerGamma/(Log[2]))-α+1/2

Pair Vacancies

Q==(QPochhammer2[(1/2), (1/2)])

Product[1-(1/(2^k)),{k,∞}]

N[%]

Product[2^i-1,{i,k}]

Q==exp(-(∑_n^∞)(1/(n ((2^n)-1))))

V8

Sum[(1/(n((2^n)-1))),{n,∞}]

N[%]

-Log[QPochhammer[(1/2),(1/2)]]//N

NSum[(1/(n((2^n)-1))),{n,∞}]

Q==(2^(13/24)) (e^(-((π^2)/(log(64))))) (Sqrt[(π/(log(2)))])(QPochhammer2[(e^(-((4 (π^2))/(log(2))))), (e^(-((4 (π^2))/(log(2)))))])

(Sqrt[((2π)/(Log[2]))])Exp[((Log[2])/24)-((π^2)/(6Log[2]))]Product[1-Exp[-((4(π^2)n)/(Log[2]))],{n,∞}]//FullSimplify

N[%]

FullSimplify[(2^(13/24)) (E^(-((π^2)/(Log[64])))) (Sqrt[(π/(Log[2]))]) QPochhammer[(E^(-((4 (π^2))/(Log[2])))),(E^(-((4 (π^2))/(Log[2]))))]-QPochhammer[(1/2),(1/2)]]

PossibleZeroQ[%]

Q==(2^(-7/24)) ((EllipticThetaPrime[1, 0, (1/(Sqrt[2]))])^(1/3))

(2^(1/24))((((1/2)EllipticThetaPrime[1,0,(2^(-1/2))]))^(1/3))//N

(2^(-7/24))((EllipticThetaPrime[1,0,(2^(-1/2))])^(1/3))//N

FullSimplify[(2^(-7/24))((EllipticThetaPrime[1,0,(2^(-1/2))])^(1/3))-QPochhammer[(1/2),(1/2)]]

FullSimplify[(2^(-7/24))((EllipticThetaPrime[1,0,(2^(-1/2))])^(1/3))]

θ==(∑_k^∞)(((k (2^((1/2) k (k-1)))) ((QPolyGamma3[0, k+1, (1/2)])-(QPolyGamma3[0, 1, (1/2)])))/(log(2) ((((-1))^k) (QPochhammer[2, 2, k]))))

Sum[(1/((2^j)-1)),{j,k}]//FullSimplify

Product[2^i-1,{i,k}]

Sum[((k (2^(k(k-1)/2)))/((((-1))^k) QPochhammer[2,2,k]))((-QPolyGamma[0,1,(1/2)]+QPolyGamma[0,1+k,(1/2)])/(Log[2])),{k,∞}]

NSum[((k (2^(k(k-1)/2)))/((((-1))^k) QPochhammer[2,2,k]))((-QPolyGamma[0,1,(1/2)]+QPolyGamma[0,1+k,(1/2)])/(Log[2])),{k,∞}]

FullSimplify[((k (2^(k(k-1)/2)))/((((-1))^k) QPochhammer[2,2,k]))((-QPolyGamma[0,1,(1/2)]+QPolyGamma[0,1+k,(1/2)])/(Log[2]))]

RealDigits[θ][[1]]//Length

one=N[1,120];

N[Sum[((k (2^(k(k-1)/2)))/(Product[2^i-one,{i,k}]))(one/((2^j)-1)),{k,#},{j,k}]&/@{300,310},110]//Timing

c==-(1/Q)((α^2)-α+(1/(log(2))))+θ+1==(1/(log(2)))(∫_0^∞)(x/((x+1) (QPochhammer2[-x, (1/2)])))dx

c=θ+1-(1/Q)((1/(Log[2]))+(α^2)-α)

(1/(Log[2]))NIntegrate[(x/(1+x))(1/(Times@@Table[1+x/2^n,{n,0,90}])),{x,0,∞},WorkingPrecision->50]

Product[1/(1+(x/(2^k))),{k,0,∞}]

(1/(Log[2]))NIntegrate[(x/(1+x))(1/(QPochhammer[-x,(1/2)])),{x,0,∞}]

Integrate[(x/(1+x))(1/(QPochhammer[-x,(1/2)])),{x,0,∞}]