A210462 -id:A210462 - cp's OEIS frontend

A075778 Decimal expansion of the real root of x^3 + x^2 - 1.

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

Offset: 0

Cino Hilliard, Oct 09 2002

Also decimal expansion of the root of x^(1/sqrt(x+1)) = (1/sqrt(x+1))^x. The root of (1/x)^(1/sqrt(x+1)) = (1/sqrt(x+1))^(1/x) is the golden ratio. - Michel Lagneau, Apr 17 2012

The following decomposition holds true: X^3 + X^2 - 1 = (X - r)*(X + i * e^(-i*a) * r^(-1/2))*(X - i * e^(i*a) * r^(-1/2)), where a = arcsin(1/(2*r^(3/2))), see A218197 for the decimal expansion of a and the paper of Witula et al. for details. - Roman Witula, Oct 22 2012

			0.7548776662466927600495088963585286918946066...

Roman Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, submitted to Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

H. R. P. Ferguson, On a Generalization of the Fibonacci Numbers Useful in Memory Allocation Schema or All About the Zeroes of Z^k - Z^{k - 1} - 1, k > 0, Fibonacci Quarterly, Volume 14, Number 3, October, 1976 (see Table 2 p. 238).
Index entries for algebraic numbers, degree 3

Cf. A060006 (inverse), A210462, A210463.

A075778 := proc()
        1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
          -% ;
end proc: # R. J. Mathar, Jan 22 2013

RealDigits[N[Solve[x^3 + x^2 - 1 == 0, x] [[1]] [[1, 2]], 111]] [[1]]
RealDigits[x /. FindRoot[x^3 + x^2 == 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Nov 23 2012 *)

PARI
```
solve(x=0, 1, x^3+x^2-1)
```

polrootsreal(x^3 + x^2 - 1)[1] \\ Charles R Greathouse IV, Jul 23 2020

Let 0 < a < 1 be any real number. Then a is the lesser and 1 is the greater and a^2/1 = 1/(a+1) and a^3 + a^2 - 1 = 0. Solving this using PARI we have 0.7548776662466927600495088964... . The general cubic can also be solved in radicals.

Equals -(1/3) + (1/3)*(25/2 - (3*sqrt(69))/2)^(1/3) + (1/3)*((1/2)*(25 + 3*sqrt(69)))^(1/3).

More terms from Robert G. Wilson v, Oct 10 2002

A078712 Series expansion of (-3 - 2*x)/(1 + x - x^3) in powers of x.

Original entry on oeis.org

-3, 1, -1, -2, 3, -4, 2, 1, -5, 7, -6, 1, 6, -12, 13, -7, -5, 18, -25, 20, -2, -23, 43, -45, 22, 21, -66, 88, -67, 1, 87, -154, 155, -68, -86, 241, -309, 223, 18, -327, 550, -532, 205, 345, -877, 1082, -737, -140, 1222, -1959, 1819, -597, -1362

Offset: 0

Ralf Stephan, Dec 19 2002

This sequence is -A001608(-n), the Perrin sequence for negative n. - T. D. Noe, Oct 10 2006

Similar to the Perrin sequence A001608, I conjecture that if p is a prime then a(p) == 1 (mod p). This implies that A001945(n) == 1 (mod p) and A001608(2*n) == 2 (mod p). - Michael Somos, Dec 25 2022

			G.f. = -3 + x - x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 2*x^6 + x^7 - 5*x^8 + 7*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Dougherty-Bliss, The Meta-C-finite Ansatz, arXiv:2206.14852 [math.CO], 2022. See page 7.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See pp. 56, 58.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1).

Cf. A001608, A001945, A210462, A210463.

I:=[-3, 1, -1]; [n le 3 select I[n] else -Self(n-1)+Self(n-3): n in [1..60]]; // Vincenzo Librandi, May 17 2013

CoefficientList[Series[(2x + 3)/(x^3 - x - 1), {x, 0, 60}], x] (* Harvey P. Dale, Mar 18 2012 *)
LinearRecurrence[{-1, 0, 1}, {-3, 1, -1}, 60] (* Harvey P. Dale, Mar 18 2012 *)
a[n_] := If[n < 0, SeriesCoefficient[(-3 + x^2)/(1 - x^2 - x^3), {x, 0, -n}], SeriesCoefficient[(-3 - 2 x)/(1 + x - x^3), {x, 0, n}]]; (* Michael Somos, Oct 15 2017 *)
Table[RootSum[-1 - # + #^3 &, #^(-n) &], {n, 0, 20}] (* Eric W. Weisstein, Jun 27 2018 *)
RootSum[-1 - # + #^3 &, #^-Range[0, 20] &] (* Eric W. Weisstein, Jun 27 2018 *)

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

{a(n) = if( n<0, polcoeff( (-3 + x^2) / (1 - x^2 - x^3) + x * O(x^-n), -n), polcoeff( (-3 - 2*x) / (1 + x - x^3) + x * O(x^n), n))}; /* Michael Somos, Oct 15 2017 */

a(n) = a(n-3) - a(n-1) with a(0)=-3, a(1)=1, a(2)=-1.
a(n) = A001945(n) - A001608(n).
a(n) ~ 2*real(r^n) with r = 0.87743... + 0.7448617...*i one inverse complex root of x^3 - x - 1 = 0 (A210462, A210463).
2*a(n) = A001608(2*n) - A001608(n)^2 follows from the Binet formula for a(n) = -p^(-n) - r^(-n) - s^(-n), where p, r, s are roots of the Perrin polynomial x^3 - x - 1. - Roman Witula, Jan 31 2013
G.f.: (2*x + 3)/(x^3 - x - 1). - Vincenzo Librandi, May 17 2013

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A210453 Decimal expansion of Sum_{n>=1} 1/(nbinomial(3n,n)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 21 2013

			0.37121697526024703447477166607535880558762946905197...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press, 2006, p. 60.

Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005.
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.

Cf. A005809, A073010, A075778, A210462, A210463.

A075778neg := proc()
        1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
end proc:
A210462 := proc()
        local a075778 ;
        a075778 := A075778neg() ;
        (1+1/a075778/(a075778-1))/2 ;
end proc:
A210463 := proc()
        local a075778,a210462 ;
        a075778 := A075778neg() ;
        a210462 := A210462() ;
        -1/a075778-a210462^2 ;
        sqrt(%) ;
end proc:
A210453 := proc()
        local v,x;
        v := 0.0 ;
        for x in [ A075778neg(), A210462()+I*A210463(), A210462()-I*A210463() ] do
                v := v+ x*log(1-1/x)/(3*x-2) ;
        end do:
        evalf(v) ;
end proc:
A210453() ;

RealDigits[ HypergeometricPFQ[{1, 1, 3/2}, {4/3, 5/3}, 4/27]/3, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

Equals Sum_{n>=1} 1/(n*A005809(n)).
Equals Integral_{x=0..1} x^2/(1-x^2+x^3) dx.
Equals Sum_(R) R*log(1-1/R)/(3*R-2) where R is summed over the set of the three constants -A075778, A210462-i*A210463 and A210462-i*A210463, i=sqrt(-1), that is, over the set of the three roots of x^3-x^2+1.
Equals (1/sqrt(23)) * (arctan(sqrt(3)/(2*phi-1)) * 18*phi/(phi^2-phi+1) - log((phi^3+1)/(phi+1)^3) * (3*sqrt(3)*phi*(1-phi))/(phi^3+1)), where phi = ((25+3*sqrt(69))/2)^(1/3) (Batir, 2005, p. 378, eq. (3.2)). - Amiram Eldar, Dec 07 2024

A210463 Decimal expansion of the absolute value of the imaginary part of the two complex roots of x^3-x^2+1.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 22 2013

An algebraic number of degree 6. - Charles R Greathouse IV, Apr 14 2014

The denominator of this algebraic number is 2, since its double is an algebraic integer. - Charles R Greathouse IV, Nov 12 2014

			0.744861766619744236593170428604392367240163...

Index entries for algebraic numbers, degree 6

Cf. A075778, A210462.

A075778neg := proc()
        1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
end proc:
A210463 := proc()
        local a075778,a210462 ;
        a075778 := A075778neg() ;
        a210462 := A210462() ;
        -1/a075778-a210462^2 ;
        sqrt(%) ;
end proc:
evalf(A210463()) ;

-((2^(1/3)*(25 - 3*Sqrt[69])^(2/3) - 2)/(2*2^(2/3)*Sqrt[3]*(25 - 3*Sqrt[69])^(1/3))) // RealDigits[#, 10, 125]& // First (* Jean-François Alcover, Feb 20 2013 *)

polrootsreal(64*x^6+32*x^4+4*x^2-23)[2] \\ Charles R Greathouse IV, Apr 14 2014

Equals sqrt(1/A075778-A210462^2).

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A075778 Decimal expansion of the real root of x^3 + x^2 - 1.

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A078712 Series expansion of (-3 - 2*x)/(1 + x - x^3) in powers of x.

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A210453 Decimal expansion of Sum_{n>=1} 1/(n*binomial(3*n,n)).

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A210463 Decimal expansion of the absolute value of the imaginary part of the two complex roots of x^3-x^2+1.

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Maple

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A210453 Decimal expansion of Sum_{n>=1} 1/(nbinomial(3n,n)).