A085846 -id:A085846 - cp's OEIS frontend

A111912 Expansion of x(2 +3x +x^2 -2x^5 -x^7 -x^8)/((1-x)(1+x)*(1-x^4+x^8)).

0, 2, 3, 3, 3, 5, 4, 6, 3, 5, 1, 5, 0, 2, -3, 1, -3, -1, -4, -2, -3, -1, -1, -1, 0, 2, 3, 3, 3, 5, 4, 6, 3, 5, 1, 5, 0, 2, -3, 1, -3, -1, -4, -2, -3, -1, -1, -1, 0, 2, 3, 3, 3, 5, 4, 6, 3, 5, 1, 5, 0, 2, -3, 1, -3, -1, -4, -2, -3, -1, -1, -1

Offset: 0

Creighton Dement, Aug 20 2005

Sequence has period 24.

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1,0,-1,0,1).

Cf. A085846, A111913, A111914, A111915.

R:=PowerSeriesRing(Integers(), 75);
Coefficients(R!( x*(2+3*x+x^2-2*x^5-x^7-x^8)/((1-x)*(1+x)*(1-x^4+x^8)) )); // G. C. Greubel, Feb 12 2021

seq(coeff(series((x*(-2-3*x-x^2+2*x^5+x^7+x^8)/((x-1)*(x+1)*(x^8-x^4+1))),x,n+1),x,n),n=0..75); # Muniru A Asiru, Jun 06 2018

LinearRecurrence[{0,1,0,1,0,-1,0,-1,0,1}, {0,2,3,3,3,5,4,6,3,5}, 75] (* G. C. Greubel, Feb 12 2021 *)

Vec(x*(-2-3*x-x^2+2*x^5+x^7+x^8)/((x-1)*(x+1)*(x^8-x^4+1))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

def A111912_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(2+3*x+x^2-2*x^5-x^7-x^8)/((1-x)*(1+x)*(1-x^4+x^8)) ).list()
A111912_list(75) # G. C. Greubel, Feb 12 2021

A111913 Expansion of x(-2-3x-x^2+x^7+x^8+2x^4) / ((x-1)(x+1)*(x^8-x^4+1)).

Original entry on oeis.org

0, 2, 3, 3, 3, 3, 6, 4, 5, 1, 5, 1, 4, -2, 1, -3, 1, -3, -2, -4, -1, -1, -1, -1, 0, 2, 3, 3, 3, 3, 6, 4, 5, 1, 5, 1, 4, -2, 1, -3, 1, -3, -2, -4, -1, -1, -1, -1, 0, 2, 3, 3, 3, 3, 6, 4, 5, 1, 5, 1, 4, -2, 1, -3, 1, -3, -2, -4, -1, -1, -1, -1, 0, 2, 3, 3, 3, 3, 6, 4, 5, 1, 5, 1, 4, -2, 1, -3, 1, -3, -2, -4, -1, -1, -1, -1, 0, 2, 3, 3

Offset: 0

Creighton Dement, Aug 20 2005

It appears that (a(n)) has period 24.
The above conjecture is correct, since x^24 = 1 mod (x-1)*(x+1)*(x^8-x^4+1). - Charles R Greathouse IV, Feb 07 2013
Floretion Algebra Multiplication Program, FAMP Code: 4ibasesigcycsumseq[ + .5'i + .5j' + .5'ij' + .5e], sumtype: Y[8] = (int)Y[6] - (int)Y[7] + Y[8] + sum (internal program code); apart from initial term 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1,0,-1,0,1).

Cf. A111912, A111914, A111915, A085846.

LinearRecurrence[{0,1,0,1,0,-1,0,-1,0,1},{0,2,3,3,3,3,6,4,5,1},120] (* Harvey P. Dale, Apr 14 2019 *)

a(n)=[0,2,3,3,3,3,6,4,5,1,5,1,4,-2,1,-3,1,-3,-2,-4,-1,-1,-1,-1][n%24+1] \\ Charles R Greathouse IV, Feb 07 2013

concat(0, Vec(x*(2 + 3*x + x^2 - 2*x^4 - x^7 - x^8) / ((1 - x)*(1 + x)*(1 - x^4 + x^8)) + O(x^80))) \\ Colin Barker, May 18 2019

a(n) = a(n-2) + a(n-4) - a(n-6) - a(n-8) + a(n-10) for n>9. - Colin Barker, May 18 2019

A111914 Expansion of -x^2(x^4-2x^3+x^2-2x+1)(x+1)^2 / ((x-1)*(x^8-x^4+1)).

Original entry on oeis.org

0, 0, 1, 1, -1, -3, -4, -4, -5, -7, -9, -9, -8, -8, -9, -9, -7, -5, -4, -4, -3, -1, 1, 1, 0, 0, 1, 1, -1, -3, -4, -4, -5, -7, -9, -9, -8, -8, -9, -9, -7, -5, -4, -4, -3, -1, 1, 1, 0, 0, 1, 1, -1, -3, -4, -4, -5, -7, -9, -9, -8, -8, -9, -9, -7, -5, -4, -4, -3, -1, 1, 1, 0, 0, 1, 1, -1, -3, -4, -4, -5, -7, -9, -9, -8, -8, -9, -9, -7, -5, -4, -4

Offset: 0

Creighton Dement, Aug 20 2005

It appears that (a(n)) has period 24.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1,0,0,-1,1).

Cf. A111912, A111913, A111915, A085846.

concat([0,0], Vec(x^2*(1 + x)^2*(1 - 2*x + x^2 - 2*x^3 + x^4) / ((1 - x)*(1 - x^4 + x^8)) + O(x^40))) \\ Colin Barker, May 18 2019

a(n) = a(n-1) + a(n-4) - a(n-5) - a(n-8) + a(n-9) for n > 8. - Colin Barker, May 18 2019

A111915 Expansion of -x^2(x-1)(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).

Original entry on oeis.org

0, 0, 1, -1, 1, -1, 2, -2, 1, -1, 1, -1, 0, 0, -1, 1, -1, 1, -2, 2, -1, 1, -1, 1, 0, 0, 1, -1, 1, -1, 2, -2, 1, -1, 1, -1, 0, 0, -1, 1, -1, 1, -2, 2, -1, 1, -1, 1, 0, 0, 1, -1, 1, -1, 2, -2, 1, -1, 1, -1, 0, 0, -1, 1, -1, 1, -2, 2, -1, 1, -1, 1, 0, 0, 1, -1, 1, -1, 2, -2, 1, -1, 1, -1, 0, 0, -1, 1, -1, 1, -2, 2, -1, 1, -1, 1, 0, 0, 1, -1

Offset: 0

Creighton Dement, Aug 20 2005

It appears that a(n) has period 24.

This is true, as (1-x^4+x^8) is the cyclotomic polynomial for n=24. - Joerg Arndt, Feb 03 2017

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,-1).

Cf. A085846, A111912, A111913, A111914.

CoefficientList[Series[-x^2*(x - 1)*(x^2 - x + 1)*(x + x^2 + 1)/(1 - x^4 + x^8), {x, 0, 100}], x] (* Wesley Ivan Hurt, Feb 03 2017 *)

a(n)=[0, 0, 1, -1, 1, -1, 2, -2, 1, -1, 1, -1, 0, 0, -1, 1, -1, 1, -2, 2, -1, 1, -1, 1][n%24+1] \\ Charles R Greathouse IV, Feb 03 2017

G.f.: -x^2*(x-1)*(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).

A169862 Decimal expansion of root of x^(x-1) = (x-1)^x.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 18 2010, based on a suggestion from Daniel Forgues

			3.2931662874118610315080282912508058643722572903271212485377103961...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Foias Constant

Equals A085846 + 1. Cf. A021002.

RealDigits[x/.FindRoot[x^(x-1)==(x-1)^x,{x,3}, WorkingPrecision->150]] [[1]] (* Harvey P. Dale, Nov 09 2011 *)

solve(x=3,4,x^(x-1)-(x-1)^x) \\ Charles R Greathouse IV, Apr 14 2014

A113554 Decimal expansion of average of e^(1/e) and Pi.

Original entry on oeis.org

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

Offset: 1

Marco Matosic, Jan 13 2006

Close to A085846 which is also close to the product Zeta(2...s) and this is itself close to 2e-Pi. The e-th root of e, eRe, is the maximum for any aRa = bRb pair. See A085846. Likewise for a^b = b^a pairs there is a minimum, e^e.

For the Foias constant F satisfying FRF = fRf, F*f is very close to the third zero of the Riemann zeta function.

			2.2931302572997796860604912459379665536278824263086794933977336283...

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

R:= RealField(100); (Pi(R) + Exp(1/Exp(1)))/2; // G. C. Greubel, Aug 31 2018

First[RealDigits[N[(E^(1/E) + Pi)/2, 100]]] (* Ryan Propper, Jul 21 2006 *)

(exp(exp(-1))+Pi)/2 \\ Charles R Greathouse IV, Mar 10 2016

N((pi+exp(exp(-1)))/2,digits=107) # Danny Rorabaugh, Mar 26 2015

Equals (Pi + e^(1/e))/2.

a(18)-a(100) from Ryan Propper, Jul 21 2006

a(99)-a(100) corrected and a(101)-a(105) added by Danny Rorabaugh, Mar 26 2015

A144184 Decimal expansion of the convergent to the recurrence x = 1/(x^(1/x)-1/x-1) for all starting values of x >= 3.

Original entry on oeis.org

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

Offset: 1

Cino Hilliard, Sep 13 2008, Sep 15 2008

1/(x^(1/x)-1/x-1) ~ pi(x), the number of prime numbers <= x. This is comparable to the well known approximation Pi(x) ~ x/(log(x)-1). As x -> infinity, pi(x) - 1/(x^(1/x)-1/x-1) -> 1/2 as x-> infinity. This was derived from my original n-th root formula 1/(x^(1/x)-1) ~ pi(x). The convergent of the recurrence x = 1/(x^(1/x)-1) = 2.293166287... is expanded in A085846 and is referred to as Foias constant. The convergents 5.507985652... and 2.293166287... are both roots of 1/(x^(1/x)-1/x-1)-x = 0. 2.293166287... is also a root of 1/(x^(1/x)-1) - x = 0.

We have here examples of functions, f(x), for which we can solve for a root by recursion of the variable x. Another simple example is the recursion x = 1/(x+1).

Eric Weisstein, Foias Constant

Cf. A085846.

RealDigits[ x /. FindRoot[ 1/(x^(1/x) - 1/x - 1) - x == 0, {x, 5}, WorkingPrecision -> 100]][[1]] (* Jean-François Alcover, Dec 20 2011 *)

g(x) = 1/(x^(1/x)-1/x-1) g2(n) = a=n;for(j=1,100,a=g(a));b=eval(Vec(Str(floor(a*10^99))));
for(j=1,100,print1(b[j]","))

The convergent used to generate this sequence, 5.50798565277317825758902..., is computed with the recurrence x = 1/(x^(1/x)-1/x-1) and can also be found by solving for the roots of 1/(x^(1/x)-1/x-1)-x = 0.

A226568 Decimal expansion of the number defined by x^x + x = 1.

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Jun 11 2013

			0.30365912702996605124501895121322265999119961876117270964417033...

Cf. A085846.

RealDigits[x/.FindRoot[x^x + x == 1, {x, 3.5}, WorkingPrecision -> 110]][[1]]

solve(x=.3,1,x^x+x-1) \\ Charles R Greathouse IV, Apr 21 2016

Equals 1/(A085846+1). - Bruno Berselli, Jun 14 2013

cp's OEIS Frontend

A111912 Expansion of x*(2 +3*x +x^2 -2*x^5 -x^7 -x^8)/((1-x)*(1+x)*(1-x^4+x^8)).

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A111913 Expansion of x*(-2-3*x-x^2+x^7+x^8+2*x^4) / ((x-1)*(x+1)*(x^8-x^4+1)).

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A111914 Expansion of -x^2*(x^4-2*x^3+x^2-2*x+1)*(x+1)^2 / ((x-1)*(x^8-x^4+1)).

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A111915 Expansion of -x^2*(x-1)*(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).

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A169862 Decimal expansion of root of x^(x-1) = (x-1)^x.

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A113554 Decimal expansion of average of e^(1/e) and Pi.

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A144184 Decimal expansion of the convergent to the recurrence x = 1/(x^(1/x)-1/x-1) for all starting values of x >= 3.

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A111912 Expansion of x(2 +3x +x^2 -2x^5 -x^7 -x^8)/((1-x)(1+x)*(1-x^4+x^8)).

A111913 Expansion of x(-2-3x-x^2+x^7+x^8+2x^4) / ((x-1)(x+1)*(x^8-x^4+1)).

A111914 Expansion of -x^2(x^4-2x^3+x^2-2x+1)(x+1)^2 / ((x-1)*(x^8-x^4+1)).

A111915 Expansion of -x^2(x-1)(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).