OFFSET
0,6
LINKS
Peter Luschny, Fibonacci meanders.
FORMULA
For n > 0 let H = floor(n/2), A = floor(H/2), R = H - 1, B = A - R/2 + 1, C = A + 1, D = A - R, J = n mod 2 and Z = if(H mod 2 = 1, (H + 1)/2, H^2*(H + 2)/16) if J = 0 else Z = if(H mod 2 = 1, 1, H*(H + 2)/4); then:
a(n) = Z*Hypergeometric([1, C, C+1, D, D-J], [B, B, B-1/2, B+1/2-J], 1/16).
EXAMPLE
Fibonacci meanders classified by maximal run length of 1s (see the link) lead to the triangle
0, 1;
1, 1, 0, 1;
2, 1, 1, 1, 0, 1;
4, 3, 2, 1, 1, 1, 0, 1;
10, 7, 4, 3, 2, 1, 1, 1, 0, 1;
24, 16, 10, 7, 4, 3, 2, 1, 1, 1, 0, 1.
MAPLE
A202411 := proc(n) local A, R, B, C, D, Z, H, J; if n = 0 then RETURN(1) fi;
H:=iquo(n, 2); A:=iquo(H, 2); R:=H-1; B:=A-R/2+1; C:=A+1; D:=A-R; J:=n mod 2; if J = 0 then Z:=`if`(H mod 2 = 1, (H+1)/2, H^2*(H+2)/16) else Z:=`if`(H mod 2 = 1, 1, H*(H+2)/4) fi; Z*hypergeom([1, C, C+1, D, D-J], [B, B, B-1/2, B+1/2-J], 1/16) end:
seq(simplify(A202411(i)), i=0..42);
MATHEMATICA
A202411[0] = 1; A202411[n_] := Module[{A, R, B, C, D, Z, H, J}, H = Quotient[n, 2]; A = Quotient[H, 2]; R = H-1; B = A - R/2 + 1; C = A+1; D = A - R; J = Mod[n, 2]; If[J == 0, Z = If[Mod[H, 2] == 1, (H+1)/2, H^2*(H + 2)/16], Z = If[Mod[H, 2] == 1, 1, H*(H+2)/4]]; Z*HypergeometricPFQ[{1, C, C + 1, D, D - J}, {B, B, B - 1/2, B + 1/2 - J}, 1/16]]; Table[A202411[n], {n, 0, 42}]
(* Jean-François Alcover, Jan 27 2014, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Jan 14 2012
STATUS
approved