A209245 Main diagonal of the triple recurrence x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) with x(i,j,k) = 1 if 0 in {i,j,k}.

1, 3, 33, 543, 10497, 220503, 4870401, 111243135, 2602452993, 61985744967, 1497148260033, 36566829737727, 901314269530113, 22385640256615743, 559574590912019457, 14065064484334380543, 355222860485671141377, 9008982166319523972903, 229325469394627488082497

Offset

0,2

Comments

Level sums are defined as the sum of x(i,j,k) with i,j,k >= 0 and i+j+k = n. This gives 3*A164039(n-1) for n>0.

Slice x(1,j,k) with j,k >= 0 of the cube begins:

1, 1, 1, 1, 1, 1, 1, 1, ... A000012

1, 3, 5, 7, 9, 11, 13, 15, ... A005408

1, 5, 11, 19, 29, 41, 55, 71, ... A028387

1, 7, 19, 39, 69, 111, 167, 239, ... A108766(k+1)

1, 9, 29, 69, 139, 251, 419, 659, ...

1, 11, 41, 111, 251, 503, 923, 1583, ...

1, 13, 55, 167, 419, 923, 1847, 3431, ...

1, 15, 71, 239, 659, 1583, 3431, 6863, ...

The main diagonal of the slice is A134760.

Links

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

Formula

a(n) = x(n,n,n) with x(i,j,k) = 1 if 0 in {i,j,k} and x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) else.

a(n) ~ 3^(3*n+1/2) / (8*Pi*n). - Vaclav Kotesovec, Sep 07 2014

Maple

a:= proc(n) option remember; `if`(n<2, 2*n+1,
      ((888-3020*n+3668*n^2-1912*n^3+364*n^4) *a(n-1)
       +3*(3*n-4)*(7*n-5)*(2*n-3)*(3*n-5) *a(n-2)) /
       ((2*n-1)*(7*n-12)*(n-1)^2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 17 2013

Mathematica

b[] = 0; b[args__] := b[args] = If[{args}[[1]] == 0, 1, Sum[b @@ Sort[ ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]];
a[n_] := b @@ Table[n, 3];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 03 2018, from Alois P. Heinz's Maple code for A210472 *)

Crossrefs

Cf. A164039, A134760, A209288.

Column k=3 of A210472. - Alois P. Heinz, Jan 23 2013

Sequence in context: A221147 A355795 A291818 * A092170 A083080 A002916

Adjacent sequences: A209242 A209243 A209244 * A209246 A209247 A209248

Keyword

nonn

Author

Jon Perry, Jan 13 2013

Status

approved