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A000667 Boustrophedon transform of all-1's sequence. 28

%I

%S 1,2,4,9,24,77,294,1309,6664,38177,243034,1701909,13001604,107601977,

%T 959021574,9157981309,93282431344,1009552482977,11568619292914,

%U 139931423833509,1781662223749884,23819069385695177

%N Boustrophedon transform of all-1's sequence.

%C 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.

%C a(n) = A227862(n, n * (n mod 2)). - _Reinhard Zumkeller_, Nov 01 2013

%C Row sums of triangle A109449. - _Reinhard Zumkeller_, Nov 04 2013

%D L. 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.

%H T. D. Noe, <a href="/A000667/b000667.txt">Table of n, a(n) for n=0..100</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>

%H J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon_transform</a>

%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>

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

%F 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

%F 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

%F 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

%e ...............1..............

%e ............1..->..2..........

%e .........4..<-.3...<-..1......

%e ......1..->.5..->..8...->..9..

%t 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 Clear[nn, A1, A2, A3, A4, A5, A6, A7, A8, A9]

%t nn = 22;

%t Clear[t, n, k];

%t t[n_, 1] = 1;

%t t[n_, k_] := t[n, k] = If[n >= k, t[n - 1, k - 1] + t[n - 1, k], 0];

%t A1 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];

%t MatrixForm[A1];

%t Clear[t, n, k];

%t t[n_, 1] = If[Or[Mod[n, 4] == 1, Mod[n, 4] == 0], 1, -1];

%t t[n_, k_] := t[n, k] = If[n >= k, t[n - 1, k - 1], 0];

%t A2 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];

%t MatrixForm[A2];

%t Clear[t, n, k];

%t t[n_, k_] :=

%t t[n, k] = If[n >= k, If[n == k, 1, If[Mod[k, 2] == 1, 0, 1]], 0];

%t A3 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];

%t MatrixForm[A3];

%t Clear[t, n, k];

%t t[n_, k_] :=

%t t[n, k] = If[n >= k, If[n == k, 1, If[Mod[k, 2] == 1, 1, 0]], 0];

%t A4 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];

%t MatrixForm[A4];

%t A5 = A1*A2*A3;

%t MatrixForm[A5];

%t A6 = A1*A2*A4;

%t MatrixForm[A6];

%t A7 = Inverse[A5];

%t MatrixForm[A7];

%t A8 = Inverse[A6];

%t MatrixForm[A8];

%t A9 = A7 + A8 - IdentityMatrix[nn];

%t MatrixForm[A9];

%t Total[Transpose[A9]] (* _Mats Granvik_, Nov 25 2013 *)

%o (Sage) # Algorithm of L. Seidel (1877)

%o def A000667_list(n) :

%o R = []; A = {-1:0, 0:0}

%o k = 0; e = 1

%o for i in range(n) :

%o Am = 1

%o A[k + e] = 0

%o e = -e

%o for j in (0..i) :

%o Am += A[k]

%o A[k] = Am

%o k += e

%o # print [A[z] for z in (-i//2..i//2)]

%o R.append(A[e*i//2])

%o return R

%o A000667_list(10) # _Peter Luschny_, Jun 02 2012

%o (Haskell)

%o a000667 n = if x == 1 then last xs else x

%o where xs@(x:_) = a227862_row n

%o -- _Reinhard Zumkeller_, Nov 01 2013

%Y Absolute value of pairwise sums of A009337.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_

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Last modified December 18 10:38 EST 2014. Contains 252154 sequences.