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A062161 Boustrophedon transform of n mod 2.

%I A062161 #41 Jan 07 2025 05:45:03
%S A062161 0,1,2,4,12,36,142,624,3192,18256,116282,814144,6219972,51475776,
%T A062161 458790022,4381112064,44625674352,482962852096,5534347077362,
%U A062161 66942218896384,852334810990332,11394877025289216,159592488559874302,2336793875186479104,35703580441464231912
%N A062161 Boustrophedon transform of n mod 2.
%H A062161 Reinhard Zumkeller, <a href="/A062161/b062161.txt">Table of n, a(n) for n = 0..400</a>
%H A062161 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%H A062161 J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon 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 A062161 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>.
%H A062161 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>.
%F A062161 a(2n) = A009747(n), a(2n+1) = A002084(n).
%F A062161 E.g.f.: (sec(x)+tan(x))*sinh(x); a(n)=(A000667(n)-A062162(n))/2. - _Paul Barry_, Jan 21 2005
%F A062161 a(n) = Sum{k, k>=0} binomial(n, 2k+1)*A000111(n-2k-1). - _Philippe Deléham_, Aug 28 2005
%F A062161 a(n) = Sum_{k=0..n} A109449(n,k) * (k mod 2). - _Reinhard Zumkeller_, Nov 03 2013 [corrected by _Jason Yuen_, Jan 07 2025]
%t A062161 With[{nn=30},CoefficientList[Series[(Sec[x]+Tan[x])Sinh[x],{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Feb 16 2013 *)
%o A062161 (Sage) # Generalized algorithm of L. Seidel (1877)
%o A062161 def A062161_list(n) :
%o A062161     R = []; A = {-1:0, 0:0}
%o A062161     k = 0; e = 1
%o A062161     for i in range(n) :
%o A062161         Am = 1 if e == -1 else 0
%o A062161         A[k + e] = 0
%o A062161         e = -e
%o A062161         for j in (0..i) :
%o A062161             Am += A[k]
%o A062161             A[k] = Am
%o A062161             k += e
%o A062161         # print [A[z] for z in (-i//2..i//2)]
%o A062161         R.append(A[e*i//2])
%o A062161     return R
%o A062161 A062161_list(10) # _Peter Luschny_, Jun 02 2012
%o A062161 (Haskell)
%o A062161 a062161 n = sum $ zipWith (*) (a109449_row n) $ cycle [0,1]
%o A062161 -- _Reinhard Zumkeller_, Nov 03 2013
%o A062161 (Python)
%o A062161 from itertools import accumulate, islice
%o A062161 def A062161_gen(): # generator of terms
%o A062161     blist, m = tuple(), 1
%o A062161     while True:
%o A062161         yield (blist := tuple(accumulate(reversed(blist),initial=(m := 1-m))))[-1]
%o A062161 A062161_list = list(islice(A062161_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A062161 Cf. A000035, A062272.
%K A062161 nonn,easy
%O A062161 0,3
%A A062161 _Frank Ellermann_, Jun 10 2001