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

0, 1, 3, 8, 25, 85, 334, 1497, 7635, 43738, 278415, 1949531, 14893000, 123254221, 1098523231, 10490117340, 106851450165, 1156403632189, 13251409502982, 160286076269309, 2040825708462175, 27283829950774822, 382127363497453243, 5595206208670390323

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..400
• Peter Luschny, An old operation on sequences: the Seidel transform.
• 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).
• 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.
• Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
• Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
• N. J. A. Sloane, Transforms.
• Wikipedia, Boustrophedon transform.
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000045, A000687, A000744, A092073, A109449.

Programs

Haskell

a000738 n = sum $ zipWith (*) (a109449_row n) a000045_list
-- Reinhard Zumkeller, Nov 03 2013

Maple

read(transforms);
with(combinat):
F:=fibonacci;
[seq(F(n),n=0..50)];
BOUS2(%);

Mathematica

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 *)

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
A000738_list = list(islice(A000738_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

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

a(n) = sum(A109449(n,k)*A000045(k): k=0..n). - Reinhard Zumkeller, Nov 03 2013

Extensions

Entry revised by N. J. A. Sloane, Mar 16 2011