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

Author

N. J. A. Sloane

Keywords

• nonn

Examples

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

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, A000738, A092073.

Programs

Haskell

a000744 n = sum $ zipWith (*) (a109449_row n) $ tail a000045_list
-- Reinhard Zumkeller, Nov 03 2013

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 *)

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

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

Extensions

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