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 1-5 of 5 results.

A329665 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps UD, HH and DU.

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 38, 72, 136, 260, 499, 958, 1847, 3572, 6917, 13422, 26097, 50808, 99049, 193354, 377857, 739148, 1447292, 2836316, 5562774, 10918180, 21444029, 42143986, 82874681, 163060540, 320996342, 632211192, 1245727488, 2455674532, 4842782497, 9554018554, 18855375593, 37224944572
Offset: 0

Views

Author

Valerie Roitner, Nov 19 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(3)=6 as one has 6 meanders of length 3, namely: UUU, UUH, UHU, UHD, HUU, HUH.

Crossrefs

Cf. A308435 (avoiding UD and DU), A329666 (avoiding UU and HH).
Cf. A329664.

Formula

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

A329667 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps UU and HH.

Original entry on oeis.org

1, 2, 3, 6, 11, 21, 42, 83, 167, 341, 697, 1437, 2983, 6211, 12996, 27304, 57528, 121601, 257759, 547652, 1166299, 2489010, 5321780, 11398972, 24456235, 52549847, 113077188, 243645011, 525630690, 1135309380, 2454863253, 5313639848, 11512892983, 24967852309
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(3)=6 since we have 6 meanders of length 3, namely UHU, UDU, UHD, UDH, HUH and HUD.

Crossrefs

Cf. A329666 (excursions with same forbidden consecutive steps).

Programs

  • PARI
    my(t='t+O('t^40)); Vec((1/2)*(1-t^3-3*t^2-sqrt(t^6+2*t^5-3*t^4-6*t^3-2*t^2+1))*(t+1)/((t^2+2*t-1)*t^2)) \\ Michel Marcus, Nov 25 2019

Formula

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

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

Views

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).

A329668 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps HH and DU.

Original entry on oeis.org

1, 2, 4, 9, 18, 38, 81, 171, 366, 787, 1693, 3661, 7938, 17240, 37540, 81892, 178907, 391483, 857769, 1881618, 4132225, 9083823, 19986954, 44014447, 97002134, 213933655, 472137851, 1042626752, 2303780392, 5093189194, 11265742842, 24930884645, 55196469010, 122255756284
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(3)=9 as one has 9 meanders of length 3, namely: UUU, UUH, UUD, UDH, UHU, UHD, HUU, HUD and HUH.

Crossrefs

Cf. A329666, which counts excursions with same restrictions.

Formula

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

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).
Showing 1-5 of 5 results.

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