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A084845 Numerators of the continued fraction n+1/(n+1/...) [n times].

%I A084845 #41 Feb 16 2025 08:32:49
%S A084845 1,5,33,305,3640,53353,927843,18674305,426938895,10928351501,
%T A084845 309601751184,9616792908241,324971855514293,11868363584907985,
%U A084845 465823816409224245,19553538801258341377,874091571490181406680
%N A084845 Numerators of the continued fraction n+1/(n+1/...) [n times].
%C A084845 The n-th term of the Lucas sequence U(n,-1). The denominator is the (n-1)-th term. Adjacent terms of the sequence U(n,-1) are relatively prime. - _T. D. Noe_, Aug 19 2004
%H A084845 Alois P. Heinz, <a href="/A084845/b084845.txt">Table of n, a(n) for n = 1..386</a>
%H A084845 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasSequence.html">Lucas Sequence</a>
%F A084845 a(n) = Sum_{k=0..floor(n/2)}* binomial(n-k, k)*n^(n-2k). - _Michel Lagneau_
%F A084845 a(n) = [x^n] 1/(1 - n*x - x^2). - _Paul D. Hanna_, Dec 27 2012
%F A084845 a(n) = (s^(n+1) - (-s)^(-n-1))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - _Vladimir Reshetnikov_, May 07 2016
%F A084845 a(n) = A117715(n+1,n). - _Alois P. Heinz_, Aug 12 2017
%e A084845 a(4) = 305 since 4+1/(4+1/(4+1/4)) = 305/72.
%p A084845 A084845 := proc(n)
%p A084845     fibonacci(n+1,n) ;
%p A084845 end proc:
%p A084845 seq(A084845(n),n=1..20) ; # _Zerinvary Lajos_, Dec 01 2006
%t A084845 myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Numerator[FromContinuedFraction[myList[n]]], {n, 1, 20}]
%t A084845 Table[s=n; Do[s=n+1/s, {n-1}]; Numerator[s], {n, 20}] (* _T. D. Noe_, Aug 19 2004 *)
%o A084845 (PARI) {a(n)=polcoeff(1/(1-n*x-x^2+x*O(x^n)),n)} \\ _Paul D. Hanna_, Dec 27 2012
%o A084845 (Python)
%o A084845 from sympy import fibonacci
%o A084845 def a117715(n, m): return 0 if n==0 else fibonacci(n, m)
%o A084845 def a(n): return a117715(n + 1, n)
%o A084845 print([a(n) for n in range(1, 31)]) # _Indranil Ghosh_, Aug 12 2017
%Y A084845 Cf. A084844 (denominators).
%Y A084845 Cf. A097690, A097691, A117715.
%K A084845 frac,nonn
%O A084845 1,2
%A A084845 _Hollie L. Buchanan II_, Jun 08 2003