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A088209 Numerators of convergents of the continued fraction with the n+1 partial quotients: [1;1,1,...(n 1's)...,1,n+1], starting with [1], [1;2], [1;1,3], [1;1,1,4], ...

%I A088209 #39 Apr 23 2023 22:25:01
%S A088209 1,3,7,14,28,53,99,181,327,584,1034,1817,3173,5511,9527,16402,28136,
%T A088209 48109,82023,139481,236631,400588,676822,1141489,1921993,3231243,
%U A088209 5424679,9095126,15230452,25475429,42566379,71052157,118489383
%N A088209 Numerators of convergents of the continued fraction with the n+1 partial quotients: [1;1,1,...(n 1's)...,1,n+1], starting with [1], [1;2], [1;1,3], [1;1,1,4], ...
%C A088209 Denominators form the Les Marvin sequence: A007502(n+1).
%H A088209 Reinhard Zumkeller, <a href="/A088209/b088209.txt">Table of n, a(n) for n = 0..1000</a>
%H A088209 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A088209 G.f.: (1+x-x^3)/(1-x-x^2)^2. [Corrected by _Georg Fischer_, Aug 16 2021]
%F A088209 a(n) = Fibonacci(n) + (n+1)*Fibonacci(n+1). - _Paul Barry_, Apr 20 2004
%F A088209 a(n) = a(n-1) + a(n-2) + Lucas(n). - _Yuchun Ji_, Apr 23 2023
%e A088209 a(3)/A007502(4) = [1;1,1,4] = 14/9.
%t A088209 f[n_] := Numerator@  FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* _Robert G. Wilson v_, Mar 04 2012 *)
%t A088209 CoefficientList[Series[(1+x-x^3)/(-1+x+x^2)^2,{x,0,40}],x] (* or *) LinearRecurrence[{2,1,-2,-1},{1,3,7,14},40] (* _Harvey P. Dale_, Jul 13 2021 *)
%o A088209 (Haskell)
%o A088209 a088209 n = a088209_list !! n
%o A088209 a088209_list = zipWith (+) a000045_list $ tail a045925_list
%o A088209 -- _Reinhard Zumkeller_, Oct 01 2012, Mar 04 2012
%o A088209 (Julia) # The function 'fibrec' is defined in A354044.
%o A088209 function A088209(n)
%o A088209     n == 0 && return BigInt(1)
%o A088209     a, b = fibrec(n)
%o A088209     a + (n + 1)*b
%o A088209 end
%o A088209 println([A088209(n) for n in 0:32]) # _Peter Luschny_, May 18 2022
%Y A088209 a(n) = A109754(n, n+2) = A101220(n, 0, n+2).
%Y A088209 Cf. A007502 (the denominators), A000045, A045925.
%K A088209 frac,nonn
%O A088209 0,2
%A A088209 _Paul D. Hanna_, Sep 23 2003