A338399 Inverse boustrophedon transform of the Fibonacci numbers.

0, 1, -1, 2, -7, 15, -78, 293, -1629, 8992, -58105, 404669, -3097456, 25617669, -228373197, 2180640110, -22212371403, 240392198791, -2754699284494, 33320193986081, -424246016043385, 5671750867032228, -79436475109286061, 1163129092965592997

Offset: 0

Author

Pontus von Brömssen, Oct 24 2020

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Boustrophedon Transform
• Wikipedia, Boustrophedon transform
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000045, A000111, A000738.

Programs

Mathematica

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 *)

Python

import sympy
def A338399(n):
  T=[]
  for k in range(n+1):
    T.append(sympy.fibonacci(k))
    T.reverse()
    for i in range(k):
      T[i+1]=T[i]-T[i+1]
  return T[-1]

Python

from itertools import accumulate, islice
from operator import sub
def A338399_gen(): # generator of terms
    blist, a, b = tuple(), 0, 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=a)))[-1]
        a, b = b, a+b
A338399_list = list(islice(A338399_gen(),20)) # Chai Wah Wu, Jun 10 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000111(n-k) * A000045(k).

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]

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