A263719 Decimal expansion of the real root r of r^3 + r - 1 = 0.

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

Offset: 0

Author

Paul D. Hanna, Oct 24 2015

Keywords

• nonn
• cons

Comments

Constant from Narayana's cows sequence: Limit A000930(n)/A000930(n+1) = r.

Reciprocal of constant described by A092526.

Examples

			0.682327803828019327369483739711048256891188581897998577803728606639896...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017. See Corollary 4.22. on p. 24.
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A000930, A025157, A076725, A092526, A274112.

Programs

Mathematica

RealDigits[ ((Sqrt[93] + 9)/18)^(1/3) - ((Sqrt[93] - 9)/18)^(1/3), 10, 100][[1]] (* G. C. Greubel, May 01 2017 *)

PARI

a(n) = my(r = (sqrt(93)/18 + 1/2)^(1/3) - (sqrt(93)/18 - 1/2)^(1/3)); floor(r*10^(n+1))%10
for(n=0,120,print1(a(n),", "))

PARI

solve(r=0, 1,  r^3 + r - 1 ) \\ Michel Marcus, Oct 25 2015

Formula

r = (sqrt(93)/18 + 1/2)^(1/3) - (sqrt(93)/18 - 1/2)^(1/3).

Constant r satisfies:

(1) 1/(1 - r*i) = (r + r^2*i) where i^2 = -1.

(2) r = real( 1/(1 - r*i) ).

(3) r = norm( 1/(1 - r*i) ).

(4) r = r^2 + r^4.

Equals 1/A092526. - Vaclav Kotesovec, Nov 27 2017