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%I #19 May 13 2021 14:22:12
%S 1,4,10,26,66,168,426,1078,2722,6862,17274,43432,109086,273734,686334,
%T 1719604,4305666,10774550,26948142,67367456,168337622,420472716,
%U 1049866442,2620488898,6538734758,16310909604,40676600026,101414764862,252787228590,629960214066
%N Number of n-step two-sided prudent self-avoiding walks.
%H Alois P. Heinz, <a href="/A191605/b191605.txt">Table of n, a(n) for n = 0..750</a>
%H Mireille Bousquet-Mélou, <a href="https://dmtcs.episciences.org/3627">Families of prudent self-avoiding walks</a>, DMTCS proc. AJ, 2008, 167-180.
%H Nathan Clisby, <a href="https://www.ams.org/journals/notices/202104/rnoti-p504.pdf">Enumerative Combinatorics of Lattice Polymers</a>, Notices AMS, 68:4 (April 2021), 504-515. See P_2(x) on page 511, but beware, the denominator has a typo.
%H Enrica Duchi, <a href="https://hal.archives-ouvertes.fr/hal-00159320">On some classes of prudent walks</a>, in: FPSAC'05, Taormina, Italy, 2005.
%F G.f.: (1/(1-2*t-2*t^2+2*t^3))*(1+t-t^3+t*(1-t)*sqrt((1-t^4)/(1-2*t-t^2))). [Clarified by _N. J. A. Sloane_, Mar 15 2021]
%e a(2) = 10: NN, NE, NW, SS, SE, WW, WN, EE, EN, ES.
%p a:= n-> coeff(series((1/(1-2*t-2*t^2+2*t^3)) *(1+t-t^3+t*(1-t) *sqrt((1-t^4) /(1-2*t-t^2))), t, n+3), t, n):
%p seq(a(n), n=0..30);
%K nonn,walk
%O 0,2
%A _Alois P. Heinz_, Jun 08 2011
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