A272874 -id:A272874

Search: a272874 -id:a272874

Displaying 1-3 of 3 results found.

page 1

A137421

Decimal expansion of growth constant in random Fibonacci sequence.

+10
5

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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)

LINKS

Table of n, a(n) for n=1..98.

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

FORMULA

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)

EXAMPLE

1.20556943040059031170202861778382342637710891959769944...

MAPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A078416, A272874. A357468.

KEYWORD

easy,nonn,cons,nice

AUTHOR

Jonathan Vos Post, Apr 16 2008

EXTENSIONS

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

More terms from Jean-François Alcover, Aug 06 2014

STATUS

approved

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.

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,9

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..65.

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

EXAMPLE

{

{ 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}

}

MATHEMATICA

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]

CROSSREFS

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

Cf. A025487. - Mats Granvik, Sep 30 2017

KEYWORD

sign,tabl

AUTHOR

Mats Granvik, Jul 24 2016

STATUS

approved

A356035

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

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..88.

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

FORMULA

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

EXAMPLE

2.2055694304005903117020286177838234263771089195976994404705522035518347903...

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A137421, A272874, A289265, A376121.

KEYWORD

nonn,cons,easy,changed

AUTHOR

Wolfdieter Lang, Aug 18 2022

STATUS

approved

page 1

Search completed in 0.007 seconds