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 1-4 of 4 results.

A366588 G.f. A(x) satisfies A(x) = 1 + x^3*(1+x)*A(x)^2.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 4, 2, 5, 15, 15, 19, 56, 84, 98, 224, 420, 552, 1002, 2022, 3069, 4983, 9801, 16577, 26455, 49049, 87945, 144287, 255112, 465244, 792012, 1369862, 2482714, 4348838, 7509580, 13439724, 23911044, 41643744, 73832632, 132039816, 232391394
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A115178, A366589.
Cf. A248100.

Programs

  • Maple
    f:= gfun:-rectoproc({(4*n + 8)*a(n) + (22 + 8*n)*a(n + 1) + (14 + 4*n)*a(n + 2) + (-8 - n)*a(n + 4) + (-8 - n)*a(n + 5) = 0,a(0)=1,a(1)=0,a(2)=0,a(3)=1,a(4)=1}, a(n),remember):
map(f, [$0..30]); # Robert Israel, Oct 14 2024
  • PARI
    a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(2*k, k)/(k+1));

Formula

G.f.: A(x) = 2 / (1+sqrt(1-4*x^3*(1+x))).
a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(2*k,k)/(k+1).
(4*n + 8)*a(n) + (22 + 8*n)*a(n + 1) + (14 + 4*n)*a(n + 2) + (-8 - n)*a(n + 4) + (-8 - n)*a(n + 5) = 0. - Robert Israel, Oct 14 2024

A346504 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x)^2 / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 3, 4, 6, 14, 28, 49, 95, 196, 386, 754, 1524, 3102, 6258, 12700, 26032, 53440, 109772, 226457, 468863, 972300, 2020274, 4208530, 8784556, 18365322, 38461110, 80682740, 169501696, 356579216, 751138916, 1584281062, 3345404514, 7072055268, 14965933024, 31702754496
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 21 2021

Keywords

Crossrefs

Cf. A023426, A023431, A025243, A090345, A248100, A257290, A346503.

Programs

  • Mathematica
    nmax = 37; A[] = 0; Do[A[x] = 1 + x + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 37}]
CoefficientList[Series[(1 - x)*(1 - Sqrt[(1 - x - 4*x^3 - 4*x^4)/(1 - x)]) / (2*x^3), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 27 2023 *)

Formula

a(0) = a(1) = 1, a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
G.f.: (1-x)*(1 - sqrt((1 - x - 4*x^3 - 4*x^4)/(1-x))) / (2*x^3). - Vaclav Kotesovec, Sep 27 2023

A366554 G.f. A(x) satisfies A(x) = 1 + x + x^4*A(x)^2.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 2, 6, 6, 2, 5, 20, 30, 20, 19, 70, 140, 140, 112, 266, 630, 840, 762, 1176, 2814, 4620, 5049, 6204, 12936, 24156, 31460, 36894, 63492, 123552, 185471, 228800, 338910, 634920, 1050686, 1411410, 1944800, 3354780, 5820256, 8513804, 11644490
Offset: 0

Views

Author

Seiichi Manyama, Oct 13 2023

Keywords

Crossrefs

Cf. A007477, A025227, A248100.
Cf. A366556, A366558.
Cf. A366589.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(k+1, n-4*k)*binomial(2*k, k)/(k+1));

Formula

G.f.: A(x) = 2*(1+x) / (1+sqrt(1-4*x^4*(1+x))).
a(n) = Sum_{k=0..floor(n/4)} binomial(k+1,n-4*k) * binomial(2*k,k)/(k+1).
a(n) = A366589(n) + A366589(n-1).

A329694 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH and DU.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 8, 12, 13, 27, 50, 64, 109, 215, 322, 504, 966, 1616, 2526, 4578, 8115, 13143, 22836, 41162, 69410, 118536, 212498, 369226, 631631, 1119755, 1977612, 3419130, 6014450, 10684128, 18689970, 32807722, 58300072, 102905556, 181031164, 321348824, 570303658, 1007402762
Offset: 0

Views

Author

Valerie Roitner, Dec 06 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 at (n,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.

Examples

			a(4)=3 since we have the following 3 excursions of length 4: UHDH, HUHD and HUDH.

Crossrefs

Cf. A248100.

Formula

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

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

Content is available under The OEIS End-User License Agreement.