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

0,4

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

Table of n, a(n) for n=0..50.

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

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

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 *)

Crossrefs

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

Sequence in context: A219460 A347433 A220010 * A220204 A035618 A220931

Adjacent sequences: A190714 A190715 A190716 * A190718 A190719 A190720

Keyword

nonn,easy

Author

Johannes W. Meijer, May 18 2011

Status

approved