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Decimal expansion of square root of 71.
+10
6
8, 4, 2, 6, 1, 4, 9, 7, 7, 3, 1, 7, 6, 3, 5, 8, 6, 3, 0, 6, 3, 4, 1, 3, 9, 9, 0, 6, 2, 0, 2, 7, 3, 6, 0, 3, 1, 6, 0, 8, 0, 0, 2, 4, 0, 1, 5, 6, 0, 7, 5, 0, 0, 1, 3, 6, 6, 7, 8, 1, 1, 1, 2, 9, 3, 2, 7, 2, 2, 5, 5, 0, 2, 7, 5, 5, 2, 0, 3, 0, 5, 5, 4, 8, 5, 6, 9, 3, 5, 3, 7, 2, 9, 8, 9, 6, 9, 6, 5
COMMENTS
Continued fraction expansion is 8 followed by {2, 2, 1, 7, 1, 2, 2, 16} repeated. - Harry J. Smith, Jun 08 2009
EXAMPLE
8.426149773176358630634139906202736031608002401560750013667811129327225...
PROG
(PARI) { default(realprecision, 20080); x=sqrt(71); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010523.txt", n, " ", d)); } \\ Harry J. Smith, Jun 08 2009
Number of terms in continued fraction for sqrt(n), excl. 2nd and higher periods.
+10
5
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
REFERENCES
H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th edition, 1999, table 1.
EXAMPLE
a(2)=2: [1,(2)+ ]; a(3)=3: [1,(1,2)+ ]; a(4)=1: [2]; a(5)=2: [2,(4)+ ].
PROG
(Python)
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
CROSSREFS
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.
Denominators of continued fraction convergents to sqrt(71).
+10
2
1, 2, 5, 7, 54, 61, 176, 413, 6784, 13981, 34746, 48727, 375835, 424562, 1224959, 2874480, 47216639, 97307758, 241832155, 339139913, 2615811546, 2954951459, 8525714464, 20006380387, 328627800656, 677261981699, 1683151764054, 2360413745753, 18206047984325
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,6960,0,0,0,0,0,0,0,-1).
FORMULA
G.f.: -(x^14 -2*x^13 +5*x^12 -7*x^11 +54*x^10 -61*x^9 +176*x^8 -413*x^7 -176*x^6 -61*x^5 -54*x^4 -7*x^3 -5*x^2 -2*x -1) / (x^16 -6960*x^8 +1). - Colin Barker, Nov 13 2013
MATHEMATICA
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 6960, 0, 0, 0, 0, 0, 0, 0, -1}, {1, 2, 5, 7, 54, 61, 176, 413, 6784, 13981, 34746, 48727, 375835, 424562, 1224959, 2874480}, 30] (* Harvey P. Dale, Apr 09 2022 *)
PROG
(Magma) I:=[1, 2, 5, 7, 54, 61, 176, 413, 6784, 13981, 34746, 48727, 375835, 424562, 1224959, 2874480]; [n le 16 select I[n] else 6960*Self(n-8)-Self(n-16): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013
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