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A000744
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Boustrophedon transform (second version) of Fibonacci numbers 1,1,2,3,...
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6
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1, 2, 5, 14, 42, 144, 563, 2526, 12877, 73778, 469616, 3288428, 25121097, 207902202, 1852961189, 17694468210, 180234349762, 1950592724756, 22352145975707, 270366543452702, 3442413745494957, 46021681757269830
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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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.: (1/10)*(sec(x)+tan(x))*((5^(1/2)+1)*exp(1/2*x*(5^(1/2)+1))+(5^(1/2)-1)*exp(1/2*x*(-5^(1/2)+1)))*5^(1/2). - Sergei N. Gladkovskii, Oct 30 2014
a(n) ~ n! * (sqrt(5) - 1 + (1+sqrt(5)) * exp(sqrt(5)*Pi/2)) * 2^(n+1) / (sqrt(5) * exp((sqrt(5)-1)*Pi/4) * Pi^(n+1)). - Vaclav Kotesovec, Jun 12 2015
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EXAMPLE
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G.f. = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 144*x^5 + 563*x^6 + 2526*x^7 + ...
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MAPLE
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read(transforms);
with(combinat):
F:=fibonacci;
[seq(F(n), n=1..50)];
BOUS2(%);
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MATHEMATICA
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s[k_] := SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, k}] k!;
b[n_, k_] := Binomial[n, k] s[n - k];
a[n_] := Sum[b[n, k] Fibonacci[k + 1], {k, 0, n}];
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PROG
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(Haskell)
a000744 n = sum $ zipWith (*) (a109449_row n) $ tail a000045_list
(Python)
from itertools import accumulate, islice
def A000744_gen(): # generator of terms
blist, a, b = tuple(), 1, 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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