A288006 - OEIS

%I #24 Aug 01 2021 03:47:44

%S 1,1,1,2,2,4,6,10,15,27,43,75,124,216,364,634,1081,1879,3229,5609,

%T 9680,16809,29077,50482,87452,151811,263201,456871,792468,1375530,

%U 2386580,4142425,7188332,12476743,21652780,37582311,65225643,113210394,196487131,341036576

%N Number of distinct fountains of n coins.

%C We regard fountains as equivalent if one can be transformed into another by symmetries.

%H Alois P. Heinz, <a href="/A288006/b288006.txt">Table of n, a(n) for n = 0..4179</a>

%F a(n) = (A005169(n) + A288005(n)) / 2.

%p g:= proc(n, i) option remember; `if`(n=0, 1,

%p       add(g(n-j, j), j=1..min(i+1, n)))

%p     end:

%p b:= proc(n, i, p) option remember; `if`(n<0, 0, `if`(n=0,

%p       `if`(p<0 and i=1, 1, 0), `if`(n=i or n=i+p, 1, 0)+

%p       `if`(i<1 and p=1, 0, b(n-2*i, i, -p))+b(n-2*(i+p), i+p, -p)))

%p     end:

%p a:= n-> (g(n, 0)+`if`(n=0, 1, b(n, 0, 1)))/2:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 02 2017

%t g[n_, i_] := g[n, i] = If[n == 0, 1,

%t      Sum[g[n-j, j], {j, 1, Min[i+1, n]}]];

%t b[n_, i_, p_] := b[n, i, p] = If[n < 0, 0, If[n == 0,

%t      If[p < 0 && i == 1, 1, 0], If[n == i || n == i+p, 1, 0] +

%t      If[i < 1 && p == 1, 0, b[n - 2i, i, -p]] + b[n - 2(i+p), i+p, -p]]];

%t a[n_] := (g[n, 0] + If[n == 0, 1, b[n, 0, 1]])/2;

%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Aug 01 2021, after _Alois P. Heinz_ *)

%Y Cf. A005169, A288005.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Sep 01 2017

%E a(33)-a(39) from _Alois P. Heinz_, Sep 02 2017