A000744 Boustrophedon transform (second version) of Fibonacci numbers 1,1,2,3,...

1, 2, 5, 14, 42, 144, 563, 2526, 12877, 73778, 469616, 3288428, 25121097, 207902202, 1852961189, 17694468210, 180234349762, 1950592724756, 22352145975707, 270366543452702, 3442413745494957, 46021681757269830

Offset

0,2

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

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*A000045(k+1). - Reinhard Zumkeller, Nov 03 2013

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

Example

G.f. = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 144*x^5 + 563*x^6 + 2526*x^7 + ...

Maple

read(transforms);
with(combinat):
F:=fibonacci;
[seq(F(n), n=1..50)];
BOUS2(%);

Mathematica

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}];
Array[a, 22, 0] (* Jean-François Alcover, Jun 01 2019 *)

Prog

(Haskell)
a000744 n = sum $ zipWith (*) (a109449_row n) $ tail a000045_list
-- Reinhard Zumkeller, Nov 03 2013
(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
A000744_list = list(islice(A000744_gen(), 40)) # Chai Wah Wu, Jun 12 2022

Crossrefs

Cf. A000045, A000687, A000738, A092073.

Sequence in context: A149878 A148332 A000751 * A047046 A063545 A061058

Adjacent sequences: A000741 A000742 A000743 * A000745 A000746 A000747

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved