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A000756
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Boustrophedon transform of sequence 1,1,0,0,0,0,...
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0
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1, 2, 3, 5, 13, 41, 157, 699, 3561, 20401, 129881, 909523, 6948269, 57504201, 512516565, 4894172027, 49851629137, 539521049441, 6182455849009, 74781598946211, 952148890494165, 12729293006112121, 178281831561868013
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OFFSET
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0,2
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REFERENCES
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Ludwig Seidel, Ueber 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.
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LINKS
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Table of n, a(n) for n=0..22.
S. N. Gladkovskii, Continued fraction expansion for function sec(x) + tan(x), arXiv:1208.2243v1 [math.HO], Aug 09 2012
S. N. Gladkovskii, On the continued fraction expansion for functions 1/sin(x) + cot(x) and sec(x) + tan(x), Nov 12 2012
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, 17A 44-54 1996 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms
Index entries for sequences related to boustrophedon transform
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FORMULA
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E.g.f.: (1+x)*( tan(x) + sec(x) ).
From Sergei N. Gladkovskii Dec 03 2012 (Start)
E.g.f.: (1+x)*( 1 + x/U(0) ); U(k)= 4k + 1 - x/(2-x/(4k + 3 + x/(2+x/U(k+1) )));(continued fraction, , 4-step).
E.g.f.: (1+x)*(1 + 2*x/(U(0) - x) ) where U(k)= 4*k + 2 - x^2/U(k+1);(continued fraction, 1-step).
(End)
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PROG
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(Sage) Algorithm of L. Seidel (1877)
def A000756_list(n) :
R = []; A = {-1:1, 0:1}; k = 0; e = 1
for i in (0..n) :
Am = 0; A[k + e] = 0; e = -e
for j in (0..i) : Am += A[k]; A[k] = Am; k += e
R.append(A[-i//2] if i%2 == 0 else A[i//2])
return R
A000756_list(22) # Peter Luschny, May 27 2012
(PARI)
x='x+O('x^66);
Vec(serlaplace((1+x)*(tan(x)+ 1/cos(x))))
/* Joerg Arndt, May 28 2012 */
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CROSSREFS
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Cf. A000111.
Sequence in context: A013013 A087362 A038560 * A192241 A093999 A042445
Adjacent sequences: A000753 A000754 A000755 * A000757 A000758 A000759
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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