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A084844
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Denominators of the continued fraction n + 1/(n + 1/...) [n times].
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14
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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
Also, the n-th term of the n-th metallic sequence (the diagonal through the array A073133, and its equivalents, which is rows formed by sequences beginning with A000045, A000129, A006190, A001076, A052918) as shown below (for n>=1):
0 1 5 26 135 [701] ... A052918. (End)
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LINKS
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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
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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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A084844 :=proc(n) combinat[fibonacci](n, n) end:
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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[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)
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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