A092073 -id:A092073 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

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

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

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

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 *)

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

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

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

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

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

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

Cf. A000045, A000687, A000738, A092073.

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

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

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 *)

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

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

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

A000687 Boustrophedon transform (first version) of Fibonacci numbers 0,1,1,2,3,5,...

Original entry on oeis.org

1, 1, 2, 6, 17, 59, 229, 1029, 5242, 30040, 191201, 1338897, 10228097, 84647981, 754437958, 7204350870, 73382899597, 794189092567, 9100736472725, 110080467183393, 1401588037032782, 18737851806495008, 262435512896178877

Offset: 0

N. J. A. Sloane and Simon Plouffe

nonn

			From _John Cerkan_, Jan 25 2017: (Start)
The array begins:
                1
             0  ->  1
          2  <- 2   <-  1
       1  -> 3  ->  5   ->  6
  17  <- 16  <- 13  <-  8   <- 2 (End)

John Cerkan, Table of n, a(n) for n = 0..482
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
Index entries for sequences related to boustrophedon transform

Cf. A000045, A000738, A092073, A000744.

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

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

Entry revised by N. J. A. Sloane, Mar 15 2011

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

Original entry on oeis.org

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

N. J. A. Sloane, Apr 01 2004

nonn

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.

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

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

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

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

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

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

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A000687 Boustrophedon transform (first version) of Fibonacci numbers 0,1,1,2,3,5,...

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

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Programs

Maple

Python

Formula