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%I #24 Oct 03 2019 15:56:09
%S 1,1,1,1,2,4,8,16,33,69,145,306,651,1398,3026,6590,14425,31720,70040,
%T 155229,345193,770002,1722487,3863274,8685608,19570860,44188976,
%U 99965361,226548082,514275345,1169255837,2662319778,6070294053,13858727891,31678845485
%N G.f. A satisfies -x+(1+x^3-x)*A+(x^4-x^2)*A^2+(x^5-x^3)*A^3-x^4*A^4 = 0.
%C Used in the enumeration of prudent self-avoiding walks.
%H Alois P. Heinz, <a href="/A177794/b177794.txt">Table of n, a(n) for n = 1..1000</a>
%H S. Gao, H. Niederhausen, <a href="http://math.fau.edu/Niederhausen/HTML/Papers/Sequences%20Arising%20From%20Prudent%20Self-Avoiding%20Walks-February%2001-2010.pdf">Sequences Arising From Prudent Self-Avoiding Walks</a>, (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).
%t m = 36; A[_] = 0;
%t Do[A[x_] = (x + A[x]^2*x^2 + A[x]^3*x^3 + A[x]^2*(-1 + A[x]^2)*x^4 - A[x]^3*x^5)/(1 - x + x^3) + O[x]^m, {m}];
%t CoefficientList[A[x]/x, x] (* _Jean-François Alcover_, Oct 03 2019 *)
%o (PARI) /* verification */
%o V177794=[1, 1, 1, 1, 2, 4, 8, 16, 33, 69, 145];
%o A=x*Ser(V177794); /* = x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + ... */
%o -x+(1+x^3-x)*A+(x^4-x^2)*A^2+(x^5-x^3)*A^3-x^4*A^4 /* = O(x^12) = "zero" */
%o /* _Joerg Arndt_, May 14 2011 */
%Y Cf. A178035.
%K nonn
%O 1,5
%A This sequence was derived by Dr. Aaron Meyerowitz and submitted by _Shanzhen Gao_, May 13 2010
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