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.

A333647 Number of nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 34, 74, 169, 397, 953, 2319, 5732, 14370, 36466, 93468, 241767, 630499, 1656372, 4380128, 11652459, 31168689, 83788315, 226272531, 613632359, 1670604607, 4564607998, 12513715526, 34412992018, 94912212872, 262484672621, 727770127583
Offset: 0

Views

Author

Alois P. Heinz, Mar 31 2020

Keywords

Comments

The maximal height in all paths of length n is floor(ceil(n/2)^2/4) = A008642(n-3) for n>2.

Links

Crossrefs

Cf. A008642, A225338, A333678, A333679, A333680, A333682.

Programs

  • Maple
    b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
      b(x-1, y+j, j), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);
  • Mathematica
    b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[
     b[x-1, y+j, j], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)

A333678 Total area under all nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

Original entry on oeis.org

0, 0, 0, 2, 7, 22, 64, 196, 574, 1762, 5379, 16378, 49380, 148892, 449004, 1353718, 4076150, 12267160, 36903433, 110979048, 333628384, 1002722482, 3013085711, 9052404522, 27192329061, 81671691634, 245271884478, 736513920180, 2211445194899, 6639545054310
Offset: 0

Views

Author

Alois P. Heinz, Apr 01 2020

Keywords

Links

Crossrefs

Cf. A333647, A333679, A333680.

Programs

  • Maple
    b:= proc(x, y, t) option remember; `if`(x=0, [1, 0],
      add((p-> p+[0, p[1]*(y+j/2)])(b(x-1, y+j, j)),
           j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..38);
  • Mathematica
    b[x_, y_, t_] := b[x, y, t] = If[x == 0, {1, 0},
     Sum[Function[p, p + {0, If[p === 0, 0, p[[1]]]*(y + j/2)}][
     b[x-1, y+j, j]], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
a[n_] := b[n, 0, 0][[2]];
a /@ Range[0, 38] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)

A333680 Total number of nodes summed over all nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

Original entry on oeis.org

1, 2, 3, 8, 20, 48, 112, 272, 666, 1690, 4367, 11436, 30147, 80248, 215550, 583456, 1588956, 4351806, 11979481, 33127440, 91982688, 256354098, 716879847, 2010919560, 5656813275, 15954441334, 45106324389, 127809023944, 362897750254, 1032389760540, 2942278599032
Offset: 0

Views

Author

Alois P. Heinz, Apr 01 2020

Keywords

Links

Crossrefs

Cf. A333647, A333678, A333679.

Programs

  • Maple
    b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
      b(x-1, y+j, j), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> (n+1)*b(n, 0$2):
seq(a(n), n=0..36);
  • Mathematica
    b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[
     b[x-1, y+j, j], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
a[n_] := (n+1) b[n, 0, 0];
a /@ Range[0, 36] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)

Formula

a(n) = (n+1) * A333647(n).
Showing 1-3 of 3 results.

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