

A008909


Join 2n points on a line with n arcs above the line; form graph with the arcs as nodes, joining 2 nodes when the arcs cross. a(n) is the number of cases in which the graph is a path.


1



1, 1, 3, 8, 21, 56, 153, 428, 1222, 3549, 10454, 31159, 93801, 284788, 871007, 2681018, 8298932, 25817395, 80674901, 253106836, 796968055, 2517706036, 7977573202, 25347126629, 80738862084, 257778971503, 824798533932, 2644335308021, 8493626448823
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OFFSET

1,3


COMMENTS

a(n+1) is the number of (finite) positive integer sequences b(1),...,b(k) with b(1) + Sum_{i=1..k1} (1+max{b(i+1)b(i), 0}) <= n.  Klaus Strassburger. [E.g., a(4)=8 since the integer sequences are 1; 2; 3; 1,1; 1,2; 2,1; 2,2; 1,1,1.]


LINKS

Table of n, a(n) for n=1..29.


FORMULA

G.f. (conjecture): 1  G(0)/(1x), where G(k)= 1  1/(1  x/(1  x/(1  x/(1  x/(x  1/G(k+1)))))); (continued fraction).  Sergei N. Gladkovskii, Jul 12 2013
G.f. (conjecture): (2*x^3x^2+2*x1+sqrt(x^4+2*x^24*x+1))/(2*x^22*x).  Michael D. Weiner, Dec 17 2019


CROSSREFS

Sequence in context: A318900 A243633 A094374 * A006835 A014318 A158495
Adjacent sequences: A008906 A008907 A008908 * A008910 A008911 A008912


KEYWORD

nonn


AUTHOR

Martin Klazar


EXTENSIONS

More terms from Klaus Strassburger (strass(AT)ddfi.uniduesseldorf.de), Sep 24 2001
More terms from Sean A. Irvine, Apr 10 2018


STATUS

approved



