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A338399 Inverse boustrophedon transform of the Fibonacci numbers.

%I A338399 #13 May 09 2024 12:06:12
%S A338399 0,1,-1,2,-7,15,-78,293,-1629,8992,-58105,404669,-3097456,25617669,
%T A338399 -228373197,2180640110,-22212371403,240392198791,-2754699284494,
%U A338399 33320193986081,-424246016043385,5671750867032228,-79436475109286061,1163129092965592997
%N A338399 Inverse boustrophedon transform of the Fibonacci numbers.
%H A338399 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BoustrophedonTransform.html">Boustrophedon Transform</a>
%H A338399 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>
%H A338399 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A338399 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000111(n-k) * A000045(k).
%F A338399 E.g.f.: (2/sqrt(5)) * exp(x/2) * sinh((sqrt(5)/2)*x) * cos(x) / (1 + sin(x)). [corrected by _Vaclav Kotesovec_, May 09 2024]
%F A338399 a(n) ~ (-1)^(n+1) * sinh(sqrt(5)*Pi/4) * 2^(n + 7/2) * n^(n + 1/2) / (sqrt(5) * Pi^(n + 1/2) * exp(n + Pi/4)). - _Vaclav Kotesovec_, May 09 2024
%t A338399 nmax = 25; Round[CoefficientList[Series[2*E^(x/2)*Sinh[Sqrt[5]*x/2]*Cos[x] / (Sqrt[5]*(1 + Sin[x])), {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, May 09 2024 *)
%o A338399 (Python)
%o A338399 import sympy
%o A338399 def A338399(n):
%o A338399   T=[]
%o A338399   for k in range(n+1):
%o A338399     T.append(sympy.fibonacci(k))
%o A338399     T.reverse()
%o A338399     for i in range(k):
%o A338399       T[i+1]=T[i]-T[i+1]
%o A338399   return T[-1]
%o A338399 (Python)
%o A338399 from itertools import accumulate, islice
%o A338399 from operator import sub
%o A338399 def A338399_gen(): # generator of terms
%o A338399     blist, a, b = tuple(), 0, 1
%o A338399     while True:
%o A338399         yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=a)))[-1]
%o A338399         a, b = b, a+b
%o A338399 A338399_list = list(islice(A338399_gen(),20)) # _Chai Wah Wu_, Jun 10 2022
%Y A338399 Cf. A000045, A000111, A000738.
%K A338399 sign
%O A338399 0,4
%A A338399 _Pontus von Brömssen_, Oct 24 2020