A000667 Boustrophedon transform of all-1's sequence.

1, 2, 4, 9, 24, 77, 294, 1309, 6664, 38177, 243034, 1701909, 13001604, 107601977, 959021574, 9157981309, 93282431344, 1009552482977, 11568619292914, 139931423833509, 1781662223749884, 23819069385695177, 333601191667149054, 4884673638115922509

Offset

0,2

Comments

Fill in a triangle, like Pascal's triangle, beginning each row with a 1 and filling in rows alternately right to left and left to right.

Row sums of triangle A109449. - Reinhard Zumkeller, Nov 04 2013

Links

Alois P. Heinz, Table of n, a(n) for n = 0..485 (first 101 terms from T. D. Noe)

C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.

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, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, Transforms.

Wikipedia, Boustrophedon transform.

Index entries for sequences related to boustrophedon transform.

Formula

E.g.f.: exp(x) * (tan(x) + sec(x)).

Limit_{n->infinity} 2*n*a(n-1)/a(n) = Pi; lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1 + 1/(n-1). - Gerald McGarvey, Aug 13 2004

a(n) = Sum_{k=0..n} binomial(n, k)*A000111(n-k). a(2*n) = A000795(n) + A009747(n), a(2*n+1) = A002084(n) + A003719(n). - Philippe Deléham, Aug 28 2005

a(n) = A227862(n, n * (n mod 2)). - Reinhard Zumkeller, Nov 01 2013

G.f.: E(0)*x/(1-x)/(1-2*x) + 1/(1-x), where E(k) = 1 - x^2*(k + 1)*(k + 2)/(x^2*(k + 1)*(k + 2) - 2*(x*(k + 2) - 1)*(x*(k + 3) - 1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014

a(n) ~ n! * exp(Pi/2) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

Example

...............1..............
............1..->..2..........
.........4..<-.3...<-..1......
......1..->.5..->..8...->..9..

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[x](Tan[x]+Sec[x]), {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Nov 28 2011 *)
t[_, 0] = 1; 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 *)

Prog

(Sage) # Algorithm of L. Seidel (1877)
def A000667_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        # print [A[z] for z in (-i//2..i//2)]
        R.append(A[e*i//2])
    return R
A000667_list(10)  # Peter Luschny, Jun 02 2012
(Haskell)
a000667 n = if x == 1 then last xs else x
            where xs@(x:_) = a227862_row n
-- Reinhard Zumkeller, Nov 01 2013
(PARI) x='x+O('x^33); Vec(serlaplace( exp(x)*(tan(x) + 1/cos(x)) ) ) \\ Joerg Arndt, Jul 30 2016
(Python)
from itertools import islice, accumulate
def A000667_gen(): # generator of terms
    blist = tuple()
    while True:
        yield (blist := tuple(accumulate(reversed(blist), initial=1)))[-1]
A000667_list = list(islice(A000667_gen(), 20)) # Chai Wah Wu, Jun 11 2022

Crossrefs

Absolute value of pairwise sums of A009337.

Column k=1 of A292975.

Cf. A000111, A000795, A009747, A002084, A003719, A109449, A227862.

Sequence in context: A091151 A093542 A301927 * A131351 A091352 A135934

Adjacent sequences: A000664 A000665 A000666 * A000668 A000669 A000670

Keyword

nonn,easy,nice

Author

N. J. A. Sloane and Simon Plouffe

Status

approved