|
| |
|
|
A000667
|
|
Boustrophedon transform of all-1's sequence.
|
|
35
|
|
|
|
1, 2, 4, 9, 24, 77, 294, 1309, 6664, 38177, 243034, 1701909, 13001604, 107601977, 959021574, 9157981309, 93282431344, 1009552482977, 11568619292914, 139931423833509, 1781662223749884, 23819069385695177, 333601191667149054, 4884673638115922509
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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.
a(n) = A227862(n, n * (n mod 2)). - Reinhard Zumkeller, Nov 01 2013
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 on 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)).
Lim 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, k>=0} binomial(n, k)*A000111(n-k). a(2n) = A000795(n) + A009747(n), a(2n+1) = A002084(n) + A003719(n). - Philippe Deléham, Aug 28 2005
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
|
|
|
CROSSREFS
|
Absolute value of pairwise sums of A009337.
Column k=1 of A292975.
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, Simon Plouffe
|
|
|
STATUS
|
approved
|
| |
|
|
