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
Examples
...............1.............. ............1..->..2.......... .........4..<-.3...<-..1...... ......1..->.5..->..8...->..9..
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.
Crossrefs
Programs
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Haskell
a000667 n = if x == 1 then last xs else x where xs@(x:_) = a227862_row n -- Reinhard Zumkeller, Nov 01 2013 -
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 *) -
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
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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
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
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