A010135 -id:A010135 - cp's OEIS frontend

A010499 Decimal expansion of square root of 45.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 6 followed by {1, 2, 2, 2, 1, 12} repeated (A010135). - Harry J. Smith, Jun 06 2009

			6.708203932499369089227521006193828706321855078834577172812691736231562...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Cf. A010135 (continued fraction), A248271 (Egyptian fractions).

Cf. A020802 (reciprocal), A002163, A001622.

RealDigits[N[Sqrt[45],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

default(realprecision, 20080); x=sqrt(45); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010499.txt", n, " ", d));  \\ Harry J. Smith, Jun 06 2009

Equals 3 * A002163. - Amiram Eldar, May 25 2023
From Andrea Pinos, Nov 01 2023: (Start)
Equals phi^4 - 1/phi^4 where phi = A001622 is the golden ratio.
Equals 2*sinh(4*log(phi)). (End)

A067280 Number of terms in continued fraction for sqrt(n), excl. 2nd and higher periods.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 5, 3, 1, 2, 3, 3, 6, 5, 3, 1, 2, 3, 7, 3, 7, 7, 5, 3, 1, 2, 3, 5, 6, 3, 9, 5, 5, 5, 3, 1, 2, 3, 3, 3, 4, 3, 11, 9, 7, 13, 5, 3, 1, 2, 3, 7, 6, 7, 5, 3, 7, 8, 7, 5, 12, 5, 3, 1, 2, 3, 11, 3, 9, 7, 9, 3, 8, 6, 5, 13, 7, 5, 5, 3, 1, 2, 3, 3, 6, 11, 3, 7, 6, 3, 9, 9, 11, 17, 5, 5, 12, 5

Offset: 1

Frank Ellermann, Feb 23 2002

			a(2)=2: [1,(2)+ ]; a(3)=3: [1,(1,2)+ ]; a(4)=1: [2]; a(5)=2: [2,(4)+ ].

H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th edition, 1999, table 1.

Related sequences: 2 : A040000, ..., 44: A040037, 48: A040041, ..., 51: A040043, 56: A040048, 60: A040052, 63: A040055, ..., 66: A040057. 68: A040059, 72: A040063, 80: A040071.
Related sequences: 45: A010135, ..., 47: A010137, 52: A010138, ..., 55: A010141, 57: A010142, ..., 59: A010144. 61: A010145, 62: A010146. 67: A010147, 69: A010148, ..., 71: A010150.
Cf. A003285.

from sympy import continued_fraction_periodic
def A067280(n): return len((a := continued_fraction_periodic(0,1,n))[:1]+(a[1] if a[1:] else [])) # Chai Wah Wu, Jun 14 2022

a(n) = A003285(n) + 1. - Andrey Zabolotskiy, Jun 23 2020

Name clarified by Michel Marcus, Jun 22 2020

A064850 Period of continued fraction for sqrt(5)*n.

Original entry on oeis.org

1, 2, 6, 2, 5, 4, 10, 4, 2, 14, 12, 4, 5, 10, 28, 8, 1, 2, 4, 14, 6, 8, 6, 4, 31, 14, 10, 12, 12, 20, 8, 20, 20, 2, 52, 2, 19, 4, 28, 24, 18, 8, 50, 12, 28, 6, 10, 4, 70, 62, 8, 18, 7, 10, 6, 8, 8, 12, 72, 20, 3, 12, 8, 36, 41, 28, 86, 2, 6, 44, 84, 2, 43, 42, 120, 4, 52, 36, 28, 44, 38

Offset: 1

R. K. Guy, Oct 26 2001

nonn

			A040002 (cfrac for n=1) has period length 1, so a(1)=1. A040015 (cfrac for n=2) has period length 2, so a(2)=2. A010135 (cfrac for n=3) has period length 6, so a(3)=6. - _R. J. Mathar_, Feb 10 2016

Table[Length[ContinuedFraction[Sqrt[5]n][[2]]],{n,90}] (* Harvey P. Dale, Apr 13 2015 *)

cp's OEIS Frontend

A010499 Decimal expansion of square root of 45.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A067280 Number of terms in continued fraction for sqrt(n), excl. 2nd and higher periods.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Python

Formula

Extensions

A064850 Period of continued fraction for sqrt(5)*n.

Views

Author

Keywords

Examples

Programs

Mathematica