A190717 Triplicated tetrahedral numbers A000292.

1, 1, 1, 4, 4, 4, 10, 10, 10, 20, 20, 20, 35, 35, 35, 56, 56, 56, 84, 84, 84, 120, 120, 120, 165, 165, 165, 220, 220, 220, 286, 286, 286, 364, 364, 364, 455, 455, 455, 560, 560, 560, 680, 680, 680, 816, 816, 816, 969, 969, 969

Offset: 0

Author

Johannes W. Meijer, May 18 2011

Keywords

• nonn
• easy

Comments

The Ca1 and Ze3 triangle sums, see A180662 for their definitions, of the triangle A159797 are linear sums of shifted versions of the triplicated tetrahedral numbers, e.g. Ca1(n) = a(n-1) + a(n-2) + 2*a(n-3) + a(n-6).

The Ca1, Ca2, Ze3 and Ze4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Crossrefs

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), this sequence (triplicated), A190718 (quadruplicated), A049347, A144677.

Programs

Maple

A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3,3) end: seq(A190717(n),n=0..50);

Mathematica

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{1,1,1,4,4,4,10,10,10,20},60] (* Harvey P. Dale, Mar 09 2018 *)

Formula

a(n) = binomial(floor(n/3)+3,3).

a(n) + a(n-1) + a(n-2) = A144677(n).

a(n) = Sum_{k=0..n} (A144677(n-k)*A049347(k)).

G.f.: 1/((x-1)^4*(x^2+x+1)^3).

Sum_{n>=0} 1/a(n) = 9/2. - Amiram Eldar, Aug 18 2022