A010141 -id:A010141 - cp's OEIS frontend

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

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

A010508 Decimal expansion of square root of 55.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 7 followed by {2, 2, 2, 14} repeated. - Harry J. Smith, Jun 06 2009

			7.416198487095662948711397440800713060979904319097501598732623264343830... - _Harry J. Smith_, Jun 06 2009

Harry J. Smith, Table of n, a(n) for n = 1..20000

Cf. A010141 Continued fraction. - Harry J. Smith, Jun 06 2009

RealDigits[N[Sqrt[55],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2011 *)

{ default(realprecision, 20080); x=sqrt(55); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010508.txt", n, " ", d)); } \\ Harry J. Smith, Jun 06 2009

A041095 Denominators of continued fraction convergents to sqrt(55).

Original entry on oeis.org

1, 2, 5, 12, 173, 358, 889, 2136, 30793, 63722, 158237, 380196, 5480981, 11342158, 28165297, 67672752, 975583825, 2018840402, 5013264629, 12045369660, 173648439869, 359342249398, 892332938665, 2144008126728, 30908446712857, 63960901552442, 158830249817741

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,178,0,0,0,-1).

Cf. A041094, A010141, A010508, A020812.

I:=[1, 2, 5, 12, 173, 358, 889, 2136]; [n le 8 select I[n] else 178*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Denominator[Convergents[Sqrt[55], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,178,0,0,0,-1},{1,2,5,12,173,358,889,2136},30] (* Harvey P. Dale, Nov 24 2022 *)

G.f.: -(x^2-2*x-1)*(x^4+6*x^2+1) / (x^8-178*x^4+1). - Colin Barker, Nov 12 2013

a(n) = 178*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 12 2013

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

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A010508 Decimal expansion of square root of 55.

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A041095 Denominators of continued fraction convergents to sqrt(55).

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