A273065 -id:A273065 - cp's OEIS frontend

A273066 Decimal expansion of the real root of x^3 - 2x + 2, negated.

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

Offset: 1

Rick L. Shepherd, May 14 2016

The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. The only other real, cubic, monic polynomial with nonzero constant term and equal coefficients and roots when ignoring the leading coefficient is x^3 + x^2 - x - 1 (per the Math Overflow link).

The other two roots of x^3 - 2*x - 2 are w1*(1 + (1/9)*sqrt(57))^(1/3) + w2*(1 - (1/9)*sqrt(57))^(1/3) = -0.88464617... + 0.58974280...*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 this is -(1/3)*sqrt(6)*(cosh((1/3)*arccosh((3/4)*sqrt(6))) - sqrt(3)*sinh((1/3)*arccosh((3/4)*sqrt(6)))*i) and its complex conjugate. - Wolfdieter Lang, Sep 13 2022

			1.7692923542386314152404094643350334926705530458988570042331061304026738...

A. J. Di Scala and O. Macia, Finiteness of Ulam Polynomials, arXiv:0904.0133 [math.AG], 2009.
R. Stanley, R. Israel et al., Math Overflow: Which polynomial's roots are its coefficients?, Sep 3 2015.
P. R. Stein, On Polynomial Equations with Coefficients Equal to Their Roots, The American Mathematical Monthly, Vol. 73, No. 3 (Mar., 1966), pp. 272-274.
Index entries for algebraic numbers, degree 3.

Cf. A273065, A273067, A357102.

RealDigits[Root[x^3-2x+2,1],10,120][[1]] (* Harvey P. Dale, May 25 2022 *)

default(realprecision, 200);
-solve(x = -1.8, -1.7, x^3 - 2*x + 2)

Equals ((9-sqrt(57))^(1/3))/(3^(2/3)) + 2/((3(9-sqrt(57)))^(1/3)) (from Wolfram Alpha).
From Wolfdieter Lang, Sep 13 2022: (Start)
Equals (1 + (1/9)*sqrt(57))^(1/3) + (2/3)*(1 + (1/9)*sqrt(57))^(-1/3) [compare with the above formula which uses the negative sqrt(57)].
Equals (1 + (1/9)*sqrt(57))^(1/3) + (1 - (1/9)*sqrt(57))^(1/3).
Equals (2/3)*sqrt(6)*cosh((1/3)*arccosh((3/4)*sqrt(6))). (End)

A376943 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 0, 2, 2, 0, 0, 4, 4, 4, 4, 0, 0, 8, 8, 8, 16, 8, 8, 8, 0, 16, 16, 16, 32, 32, 32, 32, 32, 16, 16, 48, 32, 32, 64, 64, 96, 96, 96, 96, 96, 96, 64, 128, 96, 96, 160, 128, 192, 256, 256, 256, 320, 320, 320, 320, 256, 384, 384, 320, 384, 384, 448, 576, 704, 640, 768, 896

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

In general, if d >= 1, b > 0 and g.f. = Sum_{k>=0} d^k * x^(b*k^2 + c*k) * Product_{j=1..k} (1 + x^j), then a(n) ~ r^c * (1+r) * exp(sqrt((2*log(d)^2 + 8*b*log(d)*log(r) + 4*b*(2*b+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*b*(1+r))*n)), where r is the smallest positive real root of the equation d*r^(2*b)*(1+r) = 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A273065, A333179, A376944, A376945, A376947.

nmax = 80; CoefficientList[Series[Sum[2^k * x^(k*(k+1)) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 80; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^(2*k), {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ r * (1+r) * exp(sqrt((2*log(2)^2 + 8*log(2)*log(r) + 12*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((3*r + 2)*n)), where r = ((46 - 6*sqrt(57))^(1/3) + (46 + 6*sqrt(57))^(1/3) - 2)/6 is the real root of the equation 2*r^2*(1+r) = 1 (A273065).

A376945 G.f.: Sum_{k>=0} 2^k * x^(k^2) * Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 2, 2, 0, 4, 4, 4, 4, 0, 8, 8, 8, 16, 8, 8, 8, 16, 16, 16, 32, 32, 32, 32, 32, 16, 48, 48, 32, 64, 64, 96, 96, 96, 96, 96, 96, 128, 128, 96, 160, 160, 192, 256, 256, 256, 320, 320, 320, 320, 384, 384, 384, 448, 384, 512, 576, 704, 704, 768, 896, 896, 1024, 1024, 1024

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A273065, A306734, A376943, A376944, A376948.

nmax = 80; CoefficientList[Series[Sum[2^k * x^(k^2) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 80; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^(2*k - 1), {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ (1+r) * exp(sqrt((2*log(2)^2 + 8*log(2)*log(r) + 12*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((3*r + 2)*n)), where r = ((46 - 6*sqrt(57))^(1/3) + (46 + 6*sqrt(57))^(1/3) - 2)/6 is the real root of the equation 2*r^2*(1+r) = 1 (A273065).

A273067 Decimal expansion of the constant term, which is also a root, of the cubic polynomial below.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, May 15 2016

The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. See A273066, the main entry.

			0.638896919471352622365353437840971860473585092379749349122138508505851410...

Index entries for algebraic numbers, degree 3.

Cf. A273065, A273066.

default(realprecision, 200);
A273066 = -solve(x = -1.8, -1.7, x^3 - 2*x + 2);
A273067 = A273066 - 2/A273066

polrootsreal(x^3-2*x^2+4*x-2)[1] \\ Charles R Greathouse IV, Feb 11 2025

A273066 - 2/A273066.

A357109 Decimal expansion of the real root of 2x^3 - 2x^2 - 1.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Sep 29 2022

This equals r0 + 1/3 where r0 is the real root of y^3 - (1/3)*y - 31/54.
The other roots of 2*x^3 - 2*x^2 - 1 are (w1*((31 + 3*sqrt(105))/4)^(1/3) + w2*((31 - 3*sqrt(105))/4)^(1/3))/3 = -0.4819115874... + 0.6028125753...*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 (-cosh((1/3)*arccosh(31/4)) + sqrt(3)*sinh((1/3)*arccosh(31/4))*i)/3, and its complex conjugate.

			1.29715650817742437246783022983731955553805581370396822836159443088438391495...

Index entries for algebraic numbers, degree 3.

Cf. A273065.

RealDigits[x /. FindRoot[2*x^3 - 2*x^2 - 1, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)

polrootsreal(2*x^3 - 2*x^2 - 1)[1] \\ Charles R Greathouse IV, Feb 11 2025

r = (((31 + 3*sqrt(105))/4)^(1/3) + ((31 + 3*sqrt(105))/4)^(-1/3) + 1)/3.
r = (((31 + 3*sqrt(105))/4)^(1/3) + ((31 - 3*sqrt(105))/4)^(1/3) + 1)/3.
r = (2*cosh((1/3)*arccosh(31/4))+1)/3.

A356411 Sum of powers of roots of x^3 - x^2 - x - 3.

Original entry on oeis.org

3, 1, 3, 13, 19, 41, 99, 197, 419, 913, 1923, 4093, 8755, 18617, 39651, 84533, 180035, 383521, 817155, 1740781, 3708499, 7900745, 16831587, 35857829, 76391651, 162744241, 346709379, 738628573, 1573570675, 3352327385, 7141783779

Offset: 0

Greg Dresden, Aug 05 2022

a(n) is the sum of the n-th powers of the three roots of x^3 - x^2 - x - 3. These roots are c1 = 2.130395..., c2 = -0.5651977... - i*1.0434274..., and c3 = -0.5651977... + i*1.0434274..., and so a(n) = c1^n + c2^n + c3^n. The real parts of c2 and c3 are A273065.

a(n) can also be determined by Vieta's formulas and Newton's identities. For example, a(3) by definition is c1^3 + c2^3 + c3^3, and from Newton's identities this equals e1^3 - 3*e1*e2 + 3*e3 for e1, e2, e3 the elementary symmetric polynomials of x^3 - x^2 - x - 3. From Vieta's formulas we have e1 = 1, e2 = -1, and e3 = 3, giving us e1^3 - 3*e1*e2 + 3*e3 = 1 + 3 + 9 = 13, as expected.

			For n=3, a(3) = (2.130395...)^3 + (-0.5651977... - i*1.0434274...)^3 + (-0.5651977... + i*1.0434274...)^3 = 13.

Index entries for linear recurrences with constant coefficients, signature (1,1,3).

Cf. A103143, A123102, A247594, A356463, A273065 (Re c2,c3).

LinearRecurrence[{1, 1, 3}, {3, 1, 3}, 40]

polsym(x^3 - x^2 - x - 3, 35) \\ Joerg Arndt, Aug 11 2022

a(n) = a(n-1) + a(n-2) + 3*a(n-3) with a(0)=3, a(1)=1, a(2) = 3.

G.f.: (3 - 2*x - x^2)/(1 - x - x^2 - 3*x^3).

cp's OEIS Frontend

A273066 Decimal expansion of the real root of x^3 - 2x + 2, negated.

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A376943 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)) * Product_{j=1..k} (1 + x^j).

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A376945 G.f.: Sum_{k>=0} 2^k * x^(k^2) * Product_{j=1..k} (1 + x^j).

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A273067 Decimal expansion of the constant term, which is also a root, of the cubic polynomial below.

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A357109 Decimal expansion of the real root of 2*x^3 - 2*x^2 - 1.

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A356411 Sum of powers of roots of x^3 - x^2 - x - 3.

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A376943 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)) Product_{j=1..k} (1 + x^j).

A357109 Decimal expansion of the real root of 2x^3 - 2x^2 - 1.