A190718 Quadruplicated tetrahedral numbers A000292.

1, 1, 1, 1, 4, 4, 4, 4, 10, 10, 10, 10, 20, 20, 20, 20, 35, 35, 35, 35, 56, 56, 56, 56, 84, 84, 84, 84, 120, 120, 120, 120, 165, 165, 165, 165, 220, 220, 220, 220, 286, 286, 286, 286, 364, 364, 364, 364, 455, 455, 455, 455

Offset

0,5

Comments

The Gi1 triangle sums, for the definitions of these and other triangle sums see A180662, of the triangle A159797 are linear sums of shifted versions of the quadruplicated tetrahedral numbers A000292, i.e., Gi1(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-8).

The Gi1 and Gi2 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..51.

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

Formula

a(n) = binomial(floor(n/4)+3,3).

a(n-3) + a(n-2) + a(n-1) + a(n) = A144678(n).

a(n) = +a(n-1) +3*a(n-4) -3*a(n-5) -3*a(n-8) +3*a(n-9) +a(n-12) -a(n-13).

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

Sum_{n>=0} 1/a(n) = 6. - Amiram Eldar, Aug 18 2022

Maple

A190718:= proc(n) binomial(floor(n/4)+3, 3) end:
seq(A190718(n), n=0..52);

Mathematica

LinearRecurrence[{1, 0, 0, 3, -3, 0, 0, -3, 3, 0, 0, 1, -1}, {1, 1, 1, 1, 4, 4, 4, 4, 10, 10, 10, 10, 20}, 60] (* Harvey P. Dale, Oct 20 2012 *)

Crossrefs

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), A190717 (triplicated).

Sequence in context: A053187 A013189 A295643 * A035621 A046109 A294246

Adjacent sequences: A190715 A190716 A190717 * A190719 A190720 A190721

Keyword

nonn,easy

Author

Johannes W. Meijer, May 18 2011

Status

approved