cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A329664 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UD, HH and DU.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 4, 8, 12, 21, 40, 69, 122, 227, 412, 747, 1386, 2567, 4744, 8851, 16566, 31004, 58268, 109858, 207368, 392331, 744072, 1413291, 2688822, 5124738, 9781492, 18694896, 35780444, 68566567, 131546440, 252661515, 485806614, 935017790, 1801327884, 3473467328, 6703610548
Offset: 0

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Author

Valerie Roitner, Nov 19 2019

Keywords

Comments

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

Examples

			a(4)=2 as one has 2 excursions of length 4, namely: HUHD and UHDH.

Crossrefs

Cf. A004149 (avoiding UD and DU).
Cf. A329666, A329665.

Formula

G.f.: (t+1)*(1 - t - sqrt(4*t^4 - 4*t^3 + t^2 - 2*t + 1))/(2*t^3).

A329669 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps HH and DD.

Original entry on oeis.org

1, 2, 4, 10, 23, 54, 129, 307, 733, 1757, 4213, 10115, 24315, 58481, 140741, 338890, 816304, 1966929, 4740758, 11428851, 27557585, 66458601, 160295262, 386671056, 932839439, 2250660384, 5430575647, 13104191607, 31622724351, 76314992880, 184178642468, 444513674334, 1072865869705
Offset: 0

Views

Author

Valerie Roitner, Nov 25 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 two avoiding HH and DD, namely UU, UH, UD and HU.

Crossrefs

See also A329666, which counts excursions with same restrictions.
Cf. A329667, A329665 (meanders avoiding other sets of step sequences of length 2).

Formula

G.f.: (1/2)*(-t^3 - 3*t^2 - sqrt(t^6 + 2*t^5 - 3*t^4 - 6*t^3 - 2*t^2 + 1) - 2*t + 1)/((t^3 + 3*t^2 + t - 1)*t).

A329671 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH and DD.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 12, 20, 33, 61, 109, 191, 349, 639, 1159, 2133, 3953, 7311, 13595, 25417, 47570, 89272, 168126, 317226, 599699, 1136403, 2157363, 4102113, 7813560, 14906230, 28476388, 54475340, 104347011, 200113007, 384207955, 738468129, 1420824404, 2736345674, 5274795212
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). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.

Examples

			a(4)=4 since we have 4 excursions of length 4, namely UHDH, UDUD, HUHD and HUDH.

Crossrefs

Cf. A329665, which counts meanders avoiding consecutive UU, HH and DD steps.

Formula

G.f.: (1+t)*(1-t^2-2*t^3-(1+t)*sqrt(1-2*t+t^2-4*t^3+4*t^4))/(2*t^4).
D-finite with recurrence: (n+4)*a(n) +(-n-4)*a(n-1) +(-n+2)*a(n-2) -3*n*a(n-3) +6*a(n-4) +4*(n-5)*a(n-5)=0. - R. J. Mathar, Jan 09 2020
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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