A086753 Number of distinct entries in a slice of A046816.

1, 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 16, 19, 21, 24, 25, 30, 32, 37, 40, 43, 46, 51, 56, 59, 64, 67, 75, 79, 83, 91, 93, 102, 108, 111, 119, 125, 131, 139, 147, 154, 160, 167, 175, 183, 189, 199, 206, 214, 225, 233, 243, 250, 261, 268, 279, 289, 298, 309, 317

Offset: 0

Author

Jon Perry, Jul 31 2003

Keywords

• nonn

Comments

Conjectured asymptotic: a(n) ~ (n^2 + 6*n - 12)/12. - Vladimir Reshetnikov, Dec 28 2021

Examples

			The slice for n=4 is
      1
     4 4
    6 12 6
   4 12 12 4
  1 4  6  4 1
with distinct entries 1,4,6,12, so a(4) = 4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A046816.

Programs

Maple

p:= proc(i, j, k) option remember;
      if i<0 or j<0 or k<0 or i>k or j>i then 0
    elif {i, j, k}={0} then 1
    else p(i, j, k-1) +p(i-1, j, k-1) +p(i-1, j-1, k-1)
      fi
    end:
a:= n-> nops({seq(seq(p(i, j, n), j=0..i), i=0..n)}):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 14 2012
#
seq(nops({coeffs(expand((x+y+z)^n))}), n = 0 .. 100); # César Eliud Lozada, Jul 02 2015
#
seq(nops({seq(seq(n!/(i!*j!*(n-i-j)!),j=i..(n-i)/2),i=0..n/3)}), n=0..100); # Robert Israel, Jul 02 2015

Mathematica

Table[Length @ Union @ Flatten @ Table[Table[n!/(i!*j!*(n-i-j)!), {j, i, (n-i)/2}], {i, 0, n/3}], {n, 0, 100}] (* Jean-François Alcover, Mar 19 2019, after Robert Israel *)

PARI

{ pt=vector(40,i,matrix(i,i)); pt[1][1,1]=1; pt[2][1,1]=1; pt[2][2,1]=1; pt[2][2,2]=1; pt[3][1,1]=1; pt[3][2,1]=2; pt[3][2,2]=2; pt[3][3,1]=1; pt[3][3,2]=2; pt[3][3,3]=1; for (i=4,40, for (j=2,i-1, pt[i][j,1]=pt[i-1][j-1,1]+pt[i-1][j,1]; pt[i][j,j]=pt[i][j,1]; pt[i][i,j]=pt[i][j,1] ); pt[i][1,1]=1; pt[i][i,1]=1; pt[i][i,i]=1; for(j=3,i-1, for (k=2,j-1, pt[i][j,k]=pt[i-1][j,k]+pt[i-1][j-1,k]+pt[i-1][j-1,k-1]))); pt } { makept(x)=local(xl,v,vc,uc); xl=length(x); v=vector(xl*(xl+1)/2); vc=0; for (i=1,xl, for (j=1,i,v[vc++ ]=x[i,j])); v=vecsort(v); uc=1; for (i=2,length(v),if (v[i]!=v[i-1],uc++)); print1(","uc) } for (i=1,40,makept(pt([i]))

Extensions

More terms from Alois P. Heinz, Aug 14 2012