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A000738 Boustrophedon transform (first version) of Fibonacci numbers 0,1,1,2,3,...

%I A000738 #54 Jun 11 2022 15:35:11
%S A000738 0,1,3,8,25,85,334,1497,7635,43738,278415,1949531,14893000,123254221,
%T A000738 1098523231,10490117340,106851450165,1156403632189,13251409502982,
%U A000738 160286076269309,2040825708462175,27283829950774822,382127363497453243,5595206208670390323
%N A000738 Boustrophedon transform (first version) of Fibonacci numbers 0,1,1,2,3,...
%H A000738 Reinhard Zumkeller, <a href="/A000738/b000738.txt">Table of n, a(n) for n = 0..400</a>
%H A000738 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%H A000738 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 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).
%H A000738 J. Millar, N. J. A. Sloane, and N. E. Young, <a href="https://doi.org/10.1006/jcta.1996.0087">A new operation on sequences: the Boustrophedon transform</a>, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
%H A000738 Ludwig Seidel, <a href="https://babel.hathitrust.org/cgi/pt?id=hvd.32044092897461&amp;view=1up&amp;seq=175">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the <a href="https://www.hathitrust.org/accessibility">HATHI TRUST Digital Library</a>]
%H A000738 Ludwig Seidel, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1877_0157-0187.pdf">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through <a href="https://de.wikipedia.org/wiki/ZOBODAT">ZOBODAT</a>]
%H A000738 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A000738 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>.
%H A000738 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000738 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
%F A000738 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
%F A000738 a(n) = sum(A109449(n,k)*A000045(k): k=0..n). - _Reinhard Zumkeller_, Nov 03 2013
%p A000738 read(transforms);
%p A000738 with(combinat):
%p A000738 F:=fibonacci;
%p A000738 [seq(F(n),n=0..50)];
%p A000738 BOUS2(%);
%t A000738 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 A000738 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 *)
%o A000738 (Haskell)
%o A000738 a000738 n = sum $ zipWith (*) (a109449_row n) a000045_list
%o A000738 -- _Reinhard Zumkeller_, Nov 03 2013
%o A000738 (Python)
%o A000738 from itertools import islice, accumulate
%o A000738 def A000738_gen(): # generator of terms
%o A000738     blist, a, b = tuple(), 0, 1
%o A000738     while True:
%o A000738         yield (blist := tuple(accumulate(reversed(blist),initial=a)))[-1]
%o A000738         a, b = b, a+b
%o A000738 A000738_list = list(islice(A000738_gen(),30)) # _Chai Wah Wu_, Jun 11 2022
%Y A000738 Cf. A000045, A000687, A000744, A092073, A109449.
%K A000738 nonn
%O A000738 0,3
%A A000738 _N. J. A. Sloane_
%E A000738 Entry revised by _N. J. A. Sloane_, Mar 16 2011