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A084844
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Denominators of the continued fraction n + 1/(n + 1/...) [n times].
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11
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1, 2, 10, 72, 701, 8658, 129949, 2298912, 46866034, 1082120050, 27916772489, 795910114440, 24851643870041, 843458630403298, 30918112619119426, 1217359297034666112, 51240457936070359069, 2296067756927144738850, 109127748348241605689981
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OFFSET
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1,2
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COMMENTS
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The (n-1)-th term of the Lucas sequence U(n,-1). The numerator is the n-th term. Adjacent terms of the sequence U(n,-1) are relatively prime. - T. D. Noe, Aug 19 2004
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..387
Eric Weisstein's World of Mathematics, Lucas Sequence
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FORMULA
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a(n) = (s^n - (-s)^(-n))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - Vladimir Reshetnikov, May 07 2016
a(n) = y(n,n), where y(m+2,n) = n*y(m+1,n) + y(m,n), with y(0,n)=0, y(1,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 03 2017
a(n) = A117715(n,n). - Bobby Jacobs, Aug 12 2017
a(n) = [x^n] x/(1 - n*x - x^2). - Ilya Gutkovskiy, Oct 10 2017
a(n) == 0 (mod n) for even n and 1 (mod n) for odd n. - Flávio V. Fernandes, Dec 08 2020
a(n) == 0 (mod n) for even n and 1 (mod n^2) for odd n.; see A065599. - Flávio V. Fernandes, Dec 25 2020
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EXAMPLE
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a(4) = 72 since 4 + 1/(4 + 1/(4 + 1/4)) = 305/72.
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MAPLE
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with(combinat): a:=proc(n) fibonacci(n, n) end: seq(a(n), n=1..30); # Zerinvary Lajos, Jan 03 2007
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MATHEMATICA
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myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Denominator[FromContinuedFraction[myList[n]]], {n, 1, 20}]
Table[s=n; Do[s=n+1/s, {n-1}]; Denominator[s], {n, 20}] (* T. D. Noe, Aug 19 2004 *)
Table[Fibonacci[n, n], {n, 1, 20}] (* Vladimir Reshetnikov, May 07 2016 *)
Table[DifferenceRoot[Function[{y, m}, {y[2+m]==n*y[1+m]+y[m], y[0]==0, y[1]==1}]][n], {n, 1, 20}] (* Benedict W. J. Irwin, Nov 03 2016 *)
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PROG
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(Python)
from sympy import fibonacci
def a(n): return fibonacci(n, n)
print map(a, range(1, 31)) # Indranil Ghosh, Aug 12 2017
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CROSSREFS
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Cf. A084845 (numerators).
Cf. A000045, A097690, A097691, A117715, A290864 (primes in this sequence).
Sequence in context: A185183 A052555 A204808 * A144011 A238085 A277502
Adjacent sequences: A084841 A084842 A084843 * A084845 A084846 A084847
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KEYWORD
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frac,nonn
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AUTHOR
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Hollie L. Buchanan II, Jun 08 2003
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STATUS
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approved
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