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A126931 a(n) = A127359(n+1)/2 - A127359(n). 7
1, 3, 10, 33, 110, 366, 1220, 4065, 13550, 45162, 150540, 501786, 1672620, 5575356, 18584520, 61948257, 206494190, 688313490, 2294378300, 7647926046, 25493086820, 84976950468, 283256501560, 944188318938, 3147294396460 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Hankel transform is A000012=[1,1,1,1,1,1,1,...].

a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and there are no (1,0)-steps at a higher level. Example: a(3)=33 because, denoting  U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, and 3 paths of shape UDH. - Emeric Deutsch, May 02 2011

LINKS

Table of n, a(n) for n=0..24.

FORMULA

G.f.: 1/(1-3x-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). [From Paul Barry, Mar 10 2009]

G.f. = 2/[1-6z+sqrt(1-4z^2)]. - Emeric Deutsch, May 02 2011

Conjecture: 3*(n+1)*a(n) +10*(-n-1)*a(n-1) +12*(-n+2)*a(n-2) +40*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 26 2012

MAPLE

A127359 := proc(n) add(binomial(n, floor(k/2))*3^(n-k), k=0..n) ; end proc: A126931 := proc(n) A127359(n+1)/2-A127359(n) ; end proc: seq(A126931(n), n=0..50) ; # R. J. Mathar, Mar 25 2010

CROSSREFS

Sequence in context: A020704 A289450 A113299 * A257178 A257363 A071722

Adjacent sequences:  A126928 A126929 A126930 * A126932 A126933 A126934

KEYWORD

nonn

AUTHOR

Philippe Deléham, Mar 17 2007

EXTENSIONS

More terms from R. J. Mathar, Mar 25 2010

STATUS

approved

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Last modified October 18 20:10 EDT 2018. Contains 316325 sequences. (Running on oeis4.)