A084844 - OEIS

%I #109 Jul 05 2022 11:34:25

%S 1,2,10,72,701,8658,129949,2298912,46866034,1082120050,27916772489,

%T 795910114440,24851643870041,843458630403298,30918112619119426,

%U 1217359297034666112,51240457936070359069,2296067756927144738850,109127748348241605689981

%N Denominators of the continued fraction n + 1/(n + 1/...) [n times].

%C 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

%C From _Flávio V. Fernandes_, Mar 05 2021: (Start)

%C 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):

%C    0    1    0    1     0      1  ... A000035

%C    0   [1]   1    2     3      5  ... A000045

%C    0    1   [2]   5    12     29  ... A000129

%C    0    1    3  [10]   33    109  ... A006190

%C    0    1    4   17   [72]   305  ... A001076

%C    0    1    5   26   135   [701] ... A052918. (End)

%H Alois P. Heinz, <a href="/A084844/b084844.txt">Table of n, a(n) for n = 1..387</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasSequence.html">Lucas Sequence</a>

%F a(n) = (s^n - (-s)^(-n))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - _Vladimir Reshetnikov_, May 07 2016

%F 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

%F a(n) ~ n^(n-1). - _Vaclav Kotesovec_, Jun 03 2017

%F a(n) = A117715(n,n). - _Bobby Jacobs_, Aug 12 2017

%F a(n) = [x^n] x/(1 - n*x - x^2). - _Ilya Gutkovskiy_, Oct 10 2017

%F a(n) == 0 (mod n) for even n and 1 (mod n) for odd n. - _Flávio V. Fernandes_, Dec 08 2020

%F a(n) == 0 (mod n) for even n and 1 (mod n^2) for odd n; see A065599. - _Flávio V. Fernandes_, Dec 25 2020

%F a(n) == 0 (mod 2*(n/2)^2) for even n and 1 (mod n^2) for odd n; see A129194. - _Flávio V. Fernandes_, Feb 06 2021

%e a(4) = 72 since 4 + 1/(4 + 1/(4 + 1/4)) = 305/72.

%p A084844 :=proc(n) combinat[fibonacci](n, n) end:

%p seq(A084844(n), n=1..30); # _Zerinvary Lajos_, Jan 03 2007

%t myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Denominator[FromContinuedFraction[myList[n]]], {n, 1, 20}]

%t Table[s=n; Do[s=n+1/s, {n-1}]; Denominator[s], {n, 20}] (* _T. D. Noe_, Aug 19 2004 *)

%t Table[Fibonacci[n, n], {n, 1, 20}] (* _Vladimir Reshetnikov_, May 07 2016 *)

%t 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 *)

%o (Python)

%o from sympy import fibonacci

%o def a(n):

%o     return fibonacci(n, n)

%o print([a(n) for n in range(1, 31)]) # _Indranil Ghosh_, Aug 12 2017

%Y Cf. A084845 (numerators).

%Y Cf. A000045, A097690, A097691, A117715, A290864 (primes in this sequence).

%K frac,nonn

%O 1,2

%A _Hollie L. Buchanan II_, Jun 08 2003