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<a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

(PARI) { ) default(realprecision, 20080); x=solve(x=1, 2, x^4 - x^3 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b086106.txt", n, " ", d)); } \\ _)); \\ _Iain Fox_, Oct 23 2017

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Equals (1 + (A^2 + sqrt(A^4 - 16*u*A^2 + 2*A))/A)/4 with A = sqrt(8*u + 3/2), u = (-(Bp/2)^(1/3) + (Bm/2)^(1/3)*(1 - sqrt(3)*i)/2 - 3/8)/6. , with Bp = 27 + 3*sqrt(3*283), Bm = 27 - 3*sqrt(3*283), and i = sqrt(-1). (Standard computation of a quartic.) The other (negative) real root -A230151 is obtained by using in the first formula the negative square root. The other two complex roots are obtained by replacing A by -A in these two formulas. - Wolfdieter Lang, Aug 19 2022

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Wolfdieter Lang: Thanks Jon, the period should have been a comma. Concerning my permution edit , using bad English as you remarked,  was only because I didn't see a pink box. Next time I'll tro find the pink box.

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Jon E. Schoenfield: Thanks! Until reading your comment there just now, I don’t remember ever having seen anything indicating that the notation “choose(n,m)” might, in some situations, mean something other than “binomial(n,m)”! Now I know and will be more careful about that!

Jon E. Schoenfield: But I don’t understand what you meant when you wrote “I did this permutation in order to be able to write a pink box comment”. By “this permutation”, do you mean the edit that resulted in Revision #36? If so, I don’t understand, because this draft was in proposed mode, so you or I or anyone would have been able to write a pink-box comment, with or without making any edit. But if you meant something else by “this permutation”, then I have no idea what that was. ?:-/

Jon E. Schoenfield: Is the period in “:6. with Bp =” a mistake?

Jon E. Schoenfield: About “replacing this with that” vs. “replacing this by that”: I frequently hear well-educated native speakers of English say things like “This bulb is burned out. I need to replace it with a new one.” Or “These tires are too worn. You need to replace them with new ones.” Or “My old phone was unreliable, so I replaced it with a new one.” I’ve never heard a native speaker of English, in any sentence like one of those, speak of replacing a thing “by” a new one. I think a similar situation applies to “are obtained by replacing A with -A” vs. “are obtained by replacing A by -A”.

Jon E. Schoenfield: I’m sorry, in my comment at 03:35 (insomnia time), I made a mistake. I meant to send this: Is the period in “/6. with Bp =” a mistake?

Equals (1 + (A^2 + sqrt(A^4 - 16*u*A^2 + 2*A))/A)/4 with A = sqrt(8*u + 3/2), u = (-(Bp/2)^(1/3) + (Bm/2)^(1/3)*(1 - sqrt(3)*i)/2 - 3/8)/6. with Bp = 27 + 3*sqrt(3*283), Bm = 27 - 3*sqrt(3*283), and i = sqrt(-1). (Standard computation of a quartic.) The other (negative) real root -A230151 is obtained by using in the first formula the negative square root. The other two complex roots are obtained by replacing A by -A in these two formulas A by -A. - Wolfdieter Lang, Aug 19 2022

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Wolfdieter Lang: Thanks Jon, (I did this permutation in order to be able to write a pink box comment). [By the way, you could  have a look at my comment on choose(n, m) versus binomial (n, m) in A134346 (after edition  Nr.30).]

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A. Stakhov, and B. Rozin, <a href="http://dx.doi.org/10.1016/j.chaos.2005.04.106">Theory of Binet formulas for Fibonacci and Lucas p-numbers</a>, Chaos, Solit. Fractals 27 (2006), 1162-1177.

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Jon E. Schoenfield: At the sentence "The other two complex roots are obtained by replacing in these two formulas A by -A", I think the native speakers of English whom I've met would say "The other two complex roots are obtained by replacing A with -A in these two formulas."