|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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..k-1} (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
|
|
|
FORMULA
|
G.f. (conjecture): 1 - G(0)/(1-x), 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^3-x^2+2*x-1+sqrt(x^4+2*x^2-4*x+1))/(2*x^2-2*x). - Michael D. Weiner, Dec 17 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Sep 24 2001
|
|
STATUS
|
approved
|
|
|
|