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

Author

Johannes W. Meijer, May 18 2011

Keywords

• nonn
• easy

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

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

Crossrefs

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

Programs

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 *)

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