A090550 Decimal expansion of solution to n/x = x - n for n = 5.

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

Offset: 1

Author

Felix Tubiana, Feb 05 2004

Keywords

• easy
• nonn
• cons

Comments

n/x = x - n with n = 1 gives the Golden Ratio = 1.6180339887...

Equals n + n/(n + n/(n + n/(n + ....))) for n = 5. See also A090388. - Stanislav Sykora, Jan 23 2014

Examples

			5.85410196624968454...
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10001
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. n + n/(n + n/(n + ...)): A090388 (n = 2), A090458 (n = 3), A090488 (n = 4), A092294 (n = 6), A092290 (n = 7), A090654 (n = 8), A090655 (n = 9), A090656 (n = 10). - Stanislav Sykora, Jan 23 2014

Cf. A001622, A057088.

Programs

Mathematica

RealDigits[(5 + 3 Sqrt[5])/2, 10, 120][[1]] (* Harvey P. Dale, Nov 27 2013 *)

PARI

(5 + 3*sqrt(5))/2 \\ G. C. Greubel, Jul 03 2017

Formula

n/x = x - n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 5: x = (5 + sqrt(45))/2 = 5.85410196624968454...

Equals (5 + 3*sqrt(5))/2 = 1 + 3*phi = sqrt(5)*(phi)^2, where phi is the golden ratio. - G. C. Greubel, Jul 03 2017

Equals 2*phi^3 - phi^2. - Michel Marcus, Apr 20 2020

Minimal polynomial is x^2 - 5x - 5 (this number is an algebraic integer). - Alonso del Arte, Apr 20 2020(n).

Equals lim_{n->oo} A057088(n+1)/A057088(n) = 1 + 3*phi. - Wolfdieter Lang, Nov 16 2023