A348909 -id:A348909 - cp's OEIS frontend

A348908 Decimal expansion of the positive real root of x^4 - 3*x - 6.

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

Offset: 1

Stefano Spezia, Nov 03 2021

This constant appears in the upper bounds formula of the peak sidelobe level of Rudin-Shapiro sequences.

			1.8426267135838135939675706175443422...

Tom Høholdt, Helge Elbrønd Jensen and Jørn Justesen, Aperiodic correlations and the merit factor of a class of binary sequences (Corresp.), in IEEE Transactions on Information Theory, vol. 31, no. 4, pp. 549-552, July 1985; on Research Gate.
Daniel J. Katz and Courtney M. van der Linden, Peak Sidelobe Level and Peak Crosscorrelation of Golay-Rudin-Shapiro Sequences, arXiv:2108.07318 [cs.IT], 2021. See Theorem 1.2, p. 4.
Stefano Spezia, Exact form of the constant

Cf. A020985, A020987, A348909.

First[RealDigits[N[Root[x^4-3x-6,x,2],89]]]

solve(x=0, 2, x^4 - 3*x - 6) \\ Michel Marcus, Nov 03 2021

See the formula in Links section.

A356563 Sums of powers of roots of x^3 - 2*x^2 - x - 2.

Original entry on oeis.org

3, 2, 6, 20, 50, 132, 354, 940, 2498, 6644, 17666, 46972, 124898, 332100, 883042, 2347980, 6243202, 16600468, 44140098, 117367068, 312075170, 829797604, 2206404514, 5866756972, 15599513666, 41478593332, 110290214274

Offset: 0

Greg Dresden and Hanzhang Fang, Aug 12 2022

The three roots of x^3 - 2*x^2 - x - 2 are c1=2.65896708... = A348909+1, c2=-0.32948354... + 0.80225455...*i, and c3=-0.32948354... - 0.80225455...*i.

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 = 2, e2 = -1, and e3 = 2, giving us e1^3 - 3*e1*e2 + 3*e3 = 8 + 6 + 6 = 20, as expected.

			For n=3, a(3) = (2.65896708...)^3 + (-0.32948354... + 0.80225455...*i)^3 + (-0.32948354... - 0.80225455...*i)^3 = 20.

Index entries for linear recurrences with constant coefficients, signature (2,1,2).

Cf. A077996, A348909 (c1-1).

LinearRecurrence[{2, 1, 2}, {3, 2, 6}, 30]

a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3).
G.f.: (3 - 4 x - x^2)/(1 - 2 x - x^2 - 2 x^3).
2*a(n) = 7*b(n) - b(n+1) for b(n) = A077996(n).

cp's OEIS Frontend

A348908 Decimal expansion of the positive real root of x^4 - 3*x - 6.

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