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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A329672 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps UU.

Original entry on oeis.org

1, 2, 4, 9, 20, 46, 107, 252, 599, 1435, 3460, 8389, 20437, 49996, 122758, 302401, 747114, 1850696, 4595370, 11435380, 28513149, 71225270, 178219696, 446637759, 1120946389, 2817089354, 7088656546, 17858286741, 45039810918, 113711798916, 287369435649, 726905294670, 1840328917065
Offset: 0

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Author

Valerie Roitner, Nov 26 2019

Keywords

Comments

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). A meander is a path starting at (0,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.

Examples

			a(2)=4 since we have 4 meanders of length 2 avoiding UU, namely UH, UD, HU and HH.

Crossrefs

Cf. A004148 (shifted by 1) which counts excursions avoiding consecutive UU steps. See also A329673 and A329674 which count meanders avoiding consecutive HH and DD respectively.

Formula

G.f.: -(1+t)*(1-t-3*t^2-sqrt(1-2*t-t^2-2*t^3+t^4))/(2*t^2*(1-2*t-2*t^2)).
D-finite with recurrence (n+2)*a(n) +(-3*n-5)*a(n-1) +(-3*n+2)*a(n-2) +(5*n+2)*a(n-3) +(11*n-19)*a(n-4) +(9*n-32)*a(n-5) +2*a(n-6) +2*(-n+6)*a(n-7)=0. - R. J. Mathar, Jan 25 2023

A329674 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps DD.

Original entry on oeis.org

1, 2, 5, 13, 34, 90, 240, 643, 1729, 4662, 12597, 34095, 92406, 250719, 680877, 1850457, 5032296, 13692674, 37274438, 101509476, 276535824, 753574253, 2054064713, 5600176231, 15271331416, 41651397245, 113618996429, 309979833301, 845805408448, 2308108658854, 6299205562846
Offset: 0

Views

Author

Valerie Roitner, Nov 26 2019

Keywords

Comments

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). A meander is a path starting at (0,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.

Examples

			a(2)=5 since we have 5 meanders of length 2 avoiding DD, namely UU, UH, UD, HU and HH.

Crossrefs

Cf. A004148 (shifted by 1) which counts excursions avoiding consecutive DD steps.
Cf. A329672 and A329673 which count meanders avoiding consecutive UU or HH respectively.

Formula

G.f.: -(1-3*t-t^2-sqrt(1-2*t-t^2-2*t^3+t^4))/(2*t*(1-2*t-2*t^2)).
D-finite with recurrence (n+1)*a(n) +(-4*n-1)*a(n-1) +(n-2)*a(n-2) +(4*n+1)*a(n-3) +(7*n-23)*a(n-4) +2*(n-2)*a(n-5) +2*(-n+5)*a(n-6)=0. - R. J. Mathar, Jan 25 2023
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