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A000738
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Boustrophedon transform (first version) of Fibonacci numbers 0,1,1,2,3,...
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8
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0, 1, 3, 8, 25, 85, 334, 1497, 7635, 43738, 278415, 1949531, 14893000, 123254221, 1098523231, 10490117340, 106851450165, 1156403632189, 13251409502982, 160286076269309, 2040825708462175, 27283829950774822, 382127363497453243, 5595206208670390323
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
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FORMULA
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E.g.f.: (2/sqrt(5)) * exp(x/2) * sinh((sqrt(5)/2)*x) * (sin(x)+1) / cos(x). - Alois P. Heinz, Feb 08 2011
a(n) ~ 4*(exp(sqrt(5)*Pi/2)-1) * (2*n/Pi)^(n+1/2) * exp(Pi/4-n-sqrt(5)*Pi/4) / sqrt(5). - Vaclav Kotesovec, Oct 05 2013
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MAPLE
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read(transforms);
with(combinat):
F:=fibonacci;
[seq(F(n), n=0..50)];
BOUS2(%);
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MATHEMATICA
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FullSimplify[CoefficientList[Series[(2/Sqrt[5]) * E^(x/2) * (E^(Sqrt[5]/2*x)/2 - E^(-Sqrt[5]/2*x)/2) * (Sin[x]+1) / Cos[x], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec after Alois P. Heinz, Oct 05 2013 *)
t[n_, 0] := Fibonacci[n]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)
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PROG
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(Haskell)
a000738 n = sum $ zipWith (*) (a109449_row n) a000045_list
(Python)
from itertools import islice, accumulate
def A000738_gen(): # generator of terms
blist, a, b = tuple(), 0, 1
while True:
yield (blist := tuple(accumulate(reversed(blist), initial=a)))[-1]
a, b = b, a+b
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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