0,3

Number of paths in the right half-plane from (0,0) to (n,0) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(3)=9 because we have hhh, hH, Hh, hUD, hDU, UhD, DhU, UDh and DUh. The number of such paths restricted to the first quadrant is given in A128720. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007

Other two columns of the triangle in A059317 are given in A106053 and A106050. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007

J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.

T. D. Noe, Table of n, a(n) for n=0..200

G.f.=1/sqrt[(1+z-z^2)(1-3z-z^2)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007

Rec. rel.: (n+1)a(n+1)=(2n+1)a(n)+5na(n-1)-(2n-1)a(n-2)-(n-1)a(n-3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007

a(n)=sum{k=0..floor(n/2), C(n-k,k)*A002426(n-2k)}; [From Paul Barry (pbarry(AT)wit.ie), Nov 29 2008]

r:=proc(i, j) if i=0 then 0 elif i=1 and abs(j)>0 then 0 elif i=1 and j=0 then 1 elif i>=1 then r(i-1, j)+r(i-1, j-1)+r(i-1, j+1)+r(i-2, j) else 0 fi end: seq(r(i, 0), i=1..12); # very slow (from Emeric Deutsch, Jun 06 2004)

G:=1/sqrt((1+z-z^2)*(1-3*z-z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007

a[0]:=1: a[1]:=1: a[2]:=4: a[3]:=9: for n from 3 to 26 do a[n+1]:=((2*n+1)*a[n]+5*n*a[n-1]-(2*n-1)*a[n-2]-(n-1)*a[n-3])/(n+1) end do: seq(a[n], n=0..27); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007

Cf. A128720, A106050, A106053.

Sequence in context: A000368 A094255 A069563 this_sequence A127768 A091658 A086688

Adjacent sequences: A059342 A059343 A059344 this_sequence A059346 A059347 A059348

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Jan 27 2001

More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001