A190718 - OEIS

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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

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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: