OFFSET
0,4
COMMENTS
We regard fountains as equivalent if one can be transformed into another by symmetries.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..4179
MAPLE
g:= proc(n, i) option remember; `if`(n=0, 1,
add(g(n-j, j), j=1..min(i+1, n)))
end:
b:= proc(n, i, p) option remember; `if`(n<0, 0, `if`(n=0,
`if`(p<0 and i=1, 1, 0), `if`(n=i or n=i+p, 1, 0)+
`if`(i<1 and p=1, 0, b(n-2*i, i, -p))+b(n-2*(i+p), i+p, -p)))
end:
a:= n-> (g(n, 0)+`if`(n=0, 1, b(n, 0, 1)))/2:
seq(a(n), n=0..60); # Alois P. Heinz, Sep 02 2017
MATHEMATICA
g[n_, i_] := g[n, i] = If[n == 0, 1,
Sum[g[n-j, j], {j, 1, Min[i+1, n]}]];
b[n_, i_, p_] := b[n, i, p] = If[n < 0, 0, If[n == 0,
If[p < 0 && i == 1, 1, 0], If[n == i || n == i+p, 1, 0] +
If[i < 1 && p == 1, 0, b[n - 2i, i, -p]] + b[n - 2(i+p), i+p, -p]]];
a[n_] := (g[n, 0] + If[n == 0, 1, b[n, 0, 1]])/2;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 01 2017
EXTENSIONS
a(33)-a(39) from Alois P. Heinz, Sep 02 2017
STATUS
approved