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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..300

Alois P. Heinz, Animation of A333647(9) = 169 paths with a(9) = 1690 nodes

Wikipedia, Counting lattice paths

FORMULA

a(n) = (n+1) * A333647(n).

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-Fran├žois Alcover, Apr 26 2021, after Alois P. Heinz *)

CROSSREFS

Cf. A333647, A333678, A333679.

Sequence in context: A337281 A092031 A082914 * A254533 A095341 A167123

Adjacent sequences:  A333677 A333678 A333679 * A333681 A333682 A333683

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Apr 01 2020

STATUS

approved

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Last modified May 20 09:27 EDT 2022. Contains 353871 sequences. (Running on oeis4.)