A107905 Decimal expansion of (5+sqrt(21))/2.

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

Offset

1,1

Comments

Equals r(5) where r(x)=sqrt(r(x^2-2)) as iteration count -> oo (true for all x >= 2: r(x) + 1/r(x) = x). Related: r(3)=A104457, r(4)=A019973, r(6)=A156035, r(7)=A374883. - Charles L. Hohn, Jan 02 2026

References

D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 317. [From N. J. A. Sloane, Nov 22 2009]

Links

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Emma Y. Jin and Christian M. Reidys, Asymptotic Enumeration of RNA Structures with Pseudoknots, arXiv:0706.3137 [q-bio.BM], 2007, Theorem 5, p. 15.

Index entries for algebraic numbers, degree 2.

Formula

(4.791287...)^n = A090458 * A004254(n) + A004253(n). - Gary W. Adamson, Sep 11 2023

Equals lim_{n->oo} S(n, 5)/S(n-1, 5), with the S-Chebyshev polynomial (see A049310) S(n, 5) = A004254(n+1). - Wolfdieter Lang, Nov 15 2023

c^k = A004254(k)*c - A004254(k-1) for k >= 1, where c is the present constant. - Andrea Pinos, Jul 19 2024

Minimal polynomial: x^2 - 5*x + 1. - Stefano Spezia, Jul 02 2025

Example

4.7912878474779200032940235968640042444922282883839859513036...
The zeros at 15, 16 and 17 digits after the decimal point allow for a good rational approximation. The continued fraction is [4,1,3,1,3,1,3,...] = 4 + 1/(1+ 1/(3+ 1/(1+ 1/(3+ 1/(1+ 1/(3+ 1(/1+ ...

Mathematica

RealDigits[(5+Sqrt[21])/2, 10, 120][[1]] (* Harvey P. Dale, May 02 2011 *)

Prog

(PARI) (sqrt(21)+5)/2 \\ Charles R Greathouse IV, Feb 11 2025

Crossrefs

Equals 1+A090458. - R. J. Mathar, Aug 24 2008

Cf. A004253, A004254.

Cf. A104457, A019973, A156035, A374883.

Sequence in context: A004787 A210617 A191762 * A258707 A010299 A382821

Adjacent sequences: A107902 A107903 A107904 * A107906 A107907 A107908

Keyword

cons,easy,nonn

Author

Jonathan Vos Post, Jun 22 2007

Status

approved