A272874 -id:A272874 - cp's OEIS frontend

A137421 Decimal expansion of growth constant in random Fibonacci sequence.

1, 2, 0, 5, 5, 6, 9, 4, 3, 0, 4, 0, 0, 5, 9, 0, 3, 1, 1, 7, 0, 2, 0, 2, 8, 6, 1, 7, 7, 8, 3, 8, 2, 3, 4, 2, 6, 3, 7, 7, 1, 0, 8, 9, 1, 9, 5, 9, 7, 6, 9, 9, 4, 4, 0, 4, 7, 0, 5, 5, 2, 2, 0, 3, 5, 5, 1, 8, 3, 4, 7, 9, 0, 3, 5, 9, 1, 6, 7, 4, 6, 9, 1, 7, 6, 4, 1, 8, 2, 6, 9, 5, 7, 8, 0, 5, 2, 5

Offset: 1

Jonathan Vos Post, Apr 16 2008

Real zero of x^3 + x^2 - x - 2. - Charles R Greathouse IV, May 28 2011
This is the infinite nested radical sqrt(1+sqrt(-1+sqrt(1+sqrt(-1+...)))), evaluated as the limit for an increasing (even) number of terms (an odd number of terms gives always 1) and using the main branch of the complex sqrt(z) function. This real-valued constant is in fact the unique attractor of the complex mapping M(z)=sqrt(1+sqrt(-1+z)), with its attraction domain covering the whole complex plane, excluding z = 1, the other invariant point of M(z). Closely related is A272874. - Stanislav Sykora, May 08 2016
From Wolfdieter Lang, Oct 17 2022: (Start)
This equals r0 - 1/3 where r0 is the real root of y^3 - (4/3)*y - 43/27.
The other roots of x^3 + x^2 - x - 2 are (w1*(4*(43 + 3*sqrt(177)))^(1/3) + w2*(4*(43 - 3*sqrt(177)))^(1/3) - 2)/6 = -1.1027847152... + 0.6654569511...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
Using hyperbolic functions these roots are -(1 + 2*cosh((1/3)*arccosh(43/16)) - 2*sqrt(3)*sinh((1/3)*arccosh(43/16))*i)/3, and its complex conjugate.
(End)

			1.20556943040059031170202861778382342637710891959769944...

Elise Janvresse, Benoît Rittaud and Thierry De La Rue, Growth rate for the expected value of a generalized random Fibonacci sequence, arXiv:0804.2400 [math.PR], 2008.
Benoît Rittaud, On the Average Growth of Random Fibonacci Sequences, Journal of Integer Sequences, 10 (2007), Article 07.2.4.
Benoît Rittaud, Elise Janvresse, Emmanuel Lesigne and Jean-Christophe Novelli, Quand les maths se font discrètes, Le Pommier, 2008 (ISBN 978-2-7465-0370-0). See p. 119. [Broken link]
Index entries for algebraic numbers, degree 3

Cf. A078416, A272874. A357468.

Digits := 80 ; fsolve( x^3-2*x^2-1,x,2.2..2.3)-1.0 ; # R. J. Mathar, Apr 23 2008

RealDigits[Root[x^3 + x^2 - x - 2, x, 1], 10, 98] // First (* Jean-François Alcover, Aug 06 2014 *)

real(polroots(x^3+x^2-x-2)[1]) \\ Charles R Greathouse IV, May 28 2011

polrootsreal(x^3+x^2-x-2)[1] \\ Charles R Greathouse IV, May 14 2014

In the book by Benoît Rittaud et al. it is stated that this number is cube_root(43/54+sqrt(59/108))+cube_root(43/54-sqrt(59/108))-1/3. - Eric Desbiaux, Sep 13 2008, Oct 17 2008
The largest real solution of x = sqrt(1+sqrt(-1+x)). - Stanislav Sykora, May 08 2016
From Wolfdieter Lang, Oct 17 2022: (Start)
Equals ((4*(43 + 3*sqrt(177)))^(1/3) + 16*(4*(43 + 3*sqrt(177)))^(-1/3) - 2)/6.
Equals ((4*(43 + 3*sqrt(177)))^(1/3) + (4*(43 - 3*sqrt(177)))^(1/3) - 2)/6.
Equals (4*cosh((1/3)*arccosh(43/16)) - 1)/3. (End)

More terms from R. J. Mathar, Apr 23 2008

More terms from Jean-François Alcover, Aug 06 2014

A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

Original entry on oeis.org

1, 1, -1, -1, -1, 1, -1, 0, 2, -1, 0, 0, 2, -3, 1, -1, 2, 1, -5, 4, -1, 1, -3, 5, -8, 9, -5, 1, -1, 4, -4, -5, 15, -14, 6, -1, 0, -1, 6, -17, 29, -31, 20, -7, 1, 0, 0, 2, -13, 36, -55, 50, -27, 8, -1, 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1

Offset: 0

Mats Granvik, Jul 24 2016

From Mats Granvik, Sep 30 2017: (Start)
Conjecture: The largest absolute value of the eigenvalues of these characteristic polynomials appear to have the same prime signature in the factorization of the matrix sizes N.
In other words: Let b(N) equal the sequence of the largest absolute values of the eigenvalues of the characteristic polynomials of the matrices of size N. b(N) is then a sequence of truncated eigenvalues starting:
b(N=1..infinity)
= 1.00000, 1.61803, 1.61803, 2.00000, 1.61803, 2.20557, 1.61803, 2.32472, 2.00000, 2.20557, 1.61803, 2.67170, 1.61803, 2.20557, 2.20557, 2.61803, 1.61803, 2.67170, 1.61803, 2.67170, 2.20557, 2.20557, 1.61803, 3.08032, 2.00000, 2.20557, 2.32472, 2.67170, 1.61803, 2.93796, 1.61803, 2.89055, 2.20557, 2.20557, 2.20557, 3.21878, 1.61803, 2.20557, 2.20557, 3.08032, 1.61803, 2.93796, 1.61803, 2.67170, 2.67170, 2.20557, 1.61803, 3.45341, 2.00000, 2.67170, 2.20557, 2.67170, 1.61803, 3.08032, 2.20557, 3.08032, 2.20557, 2.20557, 1.61803, 3.53392, 1.61803, 2.20557, 2.67170, ...
It then appears that for n = 1,2,3,4,5,...,infinity we have the table:
Prime signature: b(Axxxxxx(n)) = Largest abs(eigenvalue):
p^0      : b(1)          = 1.0000000000000000000000000000...
p        : b(A000040(n)) = 1.6180339887498949025257388711...
p^2      : b(A001248(n)) = 2.0000000000000000000000000000...
p*q      : b(A006881(n)) = 2.2055694304005917238953315973...
p^3      : b(A030078(n)) = 2.3247179572447480566665944934...
p^2*q    : b(A054753(n)) = 2.6716998816571604358216518448...
p^4      : b(A030514(n)) = 2.6180339887498917939012699207...
p^3*q    : b(A065036(n)) = 3.0803227214906021558249449299...
p*q*r    : b(A007304(n)) = 2.9379558827528557962693867011...
p^5      : b(A050997(n)) = 2.8905508875432590620846440288...
p^2*q^2  : b(A085986(n)) = 3.2187765853016649941764626419...
p^4*q    : b(A178739(n)) = 3.4534111136673804054453285061...
p^2*q*r  : b(A085987(n)) = 3.5339198574905377192578725953...
p^6      : b(A030516(n)) = 3.1478990357047909043330946587...
p^3*q^2  : b(A143610(n)) = 3.7022736187975437971431347250...
p^5*q    : b(A178740(n)) = 3.8016448153137023524550386355...
p^3*q*r  : b(A189975(n)) = 4.0600260453688532535920785448...
p^7      : b(A092759(n)) = 3.3935083220984414431597997463...
p^4*q^2  : b(A189988(n)) = 4.1453038440113498808159420150...
p^2*q^2*r: b(A179643(n)) = 4.2413382309993874486053755390...
p^6*q    : b(A189987(n)) = 4.1311805192254587026923218218...
p*q*r*s  : b(A046386(n)) = 3.8825338629275134572083061357...
...
b(Axxxxxx(1)) in the sequences above, is given by A025487.
(End)
First column in the coefficients of the characteristic polynomials is the Möbius function A008683.
Row sums of coefficients start: 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
Third diagonal is a signed version of A000096.
Most of the eigenvalues are equal to 1. The number of eigenvalues equal to 1 are given by A075795 for n>1.
The first three of the eigenvalues above can be calculated as nested radicals. The fourth eigenvalue 2.205569430400590... minus 1 = 1.205569430400590... is also a nested radical.

			{
{ 1},
{ 1, -1},
{-1, -1,  1},
{-1,  0,  2,  -1},
{ 0,  0,  2,  -3,  1},
{-1,  2,  1,  -5,  4,   -1},
{ 1, -3,  5,  -8,  9,   -5,   1},
{-1,  4, -4,  -5, 15,  -14,   6,  -1},
{ 0, -1,  6, -17, 29,  -31,  20,  -7,  1},
{ 0,  0,  2, -13, 36,  -55,  50, -27,  8, -1},
{ 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1}
}

Mats Granvik, Mathematica program to verify that eigenvalues determine prime signature
OEIS Wiki, Prime signatures
Eric Weisstein, Prime signature
Index to sequences related to prime signature

Cf. A051731, A008683, A000040, A001248, A006881, A030078, A030514, A054753, A000096, A001622, A272874, A075795.

Cf. A025487. - Mats Granvik, Sep 30 2017

Clear[x, AA, nn, s]; Monitor[AA = Flatten[Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; MatrixForm[A]; a = A[[1, nn]]; A[[1, nn]] = A[[nn, nn]]; A[[nn, nn]] = a; CoefficientList[CharacteristicPolynomial[A, x], x], {nn, 1, 10}]], nn]

A356035 Decimal expansion of the real root of x^3 - 2*x^2 - 1.

Original entry on oeis.org

2, 2, 0, 5, 5, 6, 9, 4, 3, 0, 4, 0, 0, 5, 9, 0, 3, 1, 1, 7, 0, 2, 0, 2, 8, 6, 1, 7, 7, 8, 3, 8, 2, 3, 4, 2, 6, 3, 7, 7, 1, 0, 8, 9, 1, 9, 5, 9, 7, 6, 9, 9, 4, 4, 0, 4, 7, 0, 5, 5, 2, 2, 0, 3, 5, 5, 1, 8, 3, 4, 7, 9, 0, 3, 5, 9, 1, 6, 7, 4, 6, 9, 1, 7, 6, 4, 1, 8

Offset: 1

Wolfdieter Lang, Aug 18 2022

This is the minimum number having the property that there are uncountably many permutation classes with the growth rate equal to that number. [Vatter] - Andrey Zabolotskiy, Dec 04 2024

			2.2055694304005903117020286177838234263771089195976994404705522035518347903...

Vincent Vatter, Small permutation classes, Proc. London Math. Soc. (3), 103 (2011), 879-921; arXiv:0712.4006 [math.CO], 2007-2016.
Wikipedia, Supersilver ratio.
Index entries for algebraic numbers, degree 3

Cf. A137421, A272874, A289265, A376121.

First[RealDigits[N[Root[#1^3-2#1^2-1 &, 1, 0], 78]]] (* Stefano Spezia, Aug 19 2022 *)

solve(x=2, 3, x^3 - 2*x^2 - 1) \\ Michel Marcus, Aug 19 2022

polrootsreal(x^3 - 2*x^2 - 1)[1] \\ Charles R Greathouse IV, Dec 04 2024

Equals ((172 + 12*sqrt(177))^(1/3)+16/(172 + 12*sqrt(177))^(1/3) + 4)/6.
Equals ((172 + 12*sqrt(177))^(1/3) + (172 - 12*sqrt(177))^(1/3)  + 4)/6.
Equals (((1/2)*(43 + 3*sqrt(3*59)))^(1/3) + ((1/2)*(43 - 3*sqrt(3*59)))^(1/3)  + 2)/3.
Equals 2*(1 + 2*cosh(log((43 + 3*sqrt(177))/16)/3))/3. - Vaclav Kotesovec, Aug 19 2022
Equals y + 2/3 where y = 1.538902... is the real root of y^3 - (4/3)*y - 43/27.
Equals 1 + A137421. - R. J. Mathar, Sep 23 2022
Equals 1/A272874. - Hugo Pfoertner, Sep 11 2024

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A137421 Decimal expansion of growth constant in random Fibonacci sequence.

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A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

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A356035 Decimal expansion of the real root of x^3 - 2*x^2 - 1.

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Mathematica

PARI

PARI

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