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A202411 Sum_{k=floor(n/4)..R} C(k,m*k-(-1)^n*(R-k))*C(k+1,m*(k+2)-(-1)^n*(R-k+1)) where m = (n+1) mod 2 and R = (n+m-3)/2 for n>0 and a(0)=1. 5
1, 0, 1, 1, 1, 2, 3, 4, 7, 10, 16, 24, 39, 58, 95, 143, 233, 354, 577, 881, 1436, 2204, 3590, 5534, 9011, 13940, 22691, 35213, 57299, 89162, 145043, 226238, 367931, 575114, 935078, 1464382, 2380405, 3734150, 6068745, 9534594, 15492702, 24374230, 39598631 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Table of n, a(n) for n=0..42.

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

a[0] = 1; a[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[a[n], {n, 0, 42}] (* Jean-Fran├žois Alcover, Jan 27 2014, translated from Maple *)

CROSSREFS

Cf. A110236, A203611.

Sequence in context: A159288 A033320 A013982 * A293161 A235648 A051449

Adjacent sequences:  A202408 A202409 A202410 * A202412 A202413 A202414

KEYWORD

nonn

AUTHOR

Peter Luschny, Jan 14 2012

STATUS

approved

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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)