Rittaud Constant

Author

Eric W. Weisstein
December 17, 2025

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Constants/RittaudConstant.nb.

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

©2025 Wolfram Research, Inc. except for portions noted otherwise

Sources

https://cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/rittaud11.pdf

Value

Roots

x/.{ToRules[Roots[x^3+x^2-x-2==0,x]]}

Select[%,#∈Reals&]

x/.{ToRules[Roots[x^3+x^2-x-2==0,x,"Cubics"->False]]}

Select[%,#∈Reals&]

Reduce

RittaudConstant=x/.ToRules[Reduce[x^3+x^2-x-2==0&&x∈Reals,x]]

InputForm[%]

OEIS

A137421

BFileSequence[A137421]//Length

FromDigits[{BFileSequence[A137421],0}]//N[#,90]&

RittaudConstant//InputForm

Root[-2-x+x^2+x^3,x,1]

Root[-2-#+#^2+#^3&,1]

RealDigits[Root[-2-#+#^2+#^3&,1],10,110][[1]]

Numerical

N[RittaudConstant,100]

Plot

Modified from Ed Pegg, Jr.
Dec 21, 2025

For https://mathworld.wolfram.com/RandomFibonacciSequence.html  

The related mean value is called the Rittaud constant, the real value of $x^3 =2 x^2 +1$, about $2.2055694304005903117$.

RandomFibonacciPath[n_Integer?Positive,g1_:1,g2_:1]:=Module[
{
g=ConstantArray[0,n],
s
},
g[[1]]=g1;
If[n>=2,g[[2]]=g2];
Do[
s=RandomChoice[{-1,1}];
g[[k]]=Abs[g[[k-1]]+s g[[k-2]]],{k,3,n}];
g
]

RittaudMeanSequence[n_Integer?Positive,nsim_Integer?Positive]:=Module[
{
paths=Table[RandomFibonacciPath[n],{nsim}]
},
Mean/@Transpose[paths]
]

RittaudRatioData[n_Integer?Positive,nsim_Integer?Positive]:=Module[
{
means=RittaudMeanSequence[n,nsim]
},
Rest[means]/Most[means]
]

RittaudRatioPlot[n_Integer:40,nsim_Integer:2000]:=Module[
{
ratios=RittaudRatioData[n,nsim],
c=N[RittaudConstant,20]
},
ListLinePlot[
ratios,
PlotRange->All,
PlotStyle->Red,
AxesLabel->{n,Subscript[Style["m",Italic],Row[{Style["n",Italic],"+","1"}," "]]/Style[Subscript["m","n"],Italic]},
Epilog->{Dashed,Line[{{0,c},{100,c}}]},
BaseStyle->FontFamily->"Times"
]
]

RittaudRatioPlot[60,10000]