A092090 Boustrophedon transform of Fibonacci numbers 1, 2, 3, 5, 8, ...

1, 3, 8, 22, 67, 229, 897, 4023, 20512, 117516, 748031, 5237959, 40014097, 331156423, 2951484420, 28184585550, 287085799927, 3106996356945, 35603555478689, 430652619722011, 5483239453957132, 73305511708044652, 1026690239891085363, 15033060056592047307

Offset: 0

Author

N. J. A. Sloane, Apr 01 2004

Keywords

• nonn

Links

• C. A. Church and M. Bicknell, Exponential generating functions for Fibonacci identities, Fibonacci Quarterly, 11(3) (1973), 275-281.
• J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (Abstract, pdf, ps).
• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A000744 (which uses BOUS2), A062122 (which uses Fibonacci numbers with an error in them), A092073.

Programs

Maple

read transforms; with(combinat, fibonacci): a := [seq(fibonacci(i),i=2..30)]: BOUS2(a);

Python

from itertools import accumulate, islice
def A092090_gen(): # generator of terms
    blist, a, b = tuple(), 1, 2
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=a)))[-1]
        a, b = b, a+b
A092090_list = list(islice(A092090_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

E.g.f.: (sec(x) + tan(x))*(a^2*exp(a*x) - b^2*exp(b*x))/(a - b), where a = (1 + sqrt(5))/2 and b = (1 - sqrt(5))/2. - Petros Hadjicostas, Feb 16 2021