A211534 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w = 3x + 3y.

0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 3, 3, 6, 6, 6, 10, 10, 10, 15, 15, 15, 21, 21, 21, 28, 28, 28, 36, 36, 36, 45, 45, 45, 55, 55, 55, 66, 66, 66, 78, 78, 78, 91, 91, 91, 105, 105, 105, 120, 120, 120, 136, 136, 136, 153, 153, 153, 171, 171, 171, 190, 190, 190, 210

Offset: 0

Author

Clark Kimberling, Apr 15 2012

Keywords

• nonn
• easy

Comments

This sequence consists of six 0's followed by triply repeated triangular numbers.

For a guide to related sequences, see A211422.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A211422, A008805 (w = 2x + 2y and doubly repeated triangular numbers).

Programs

Magma

[Floor(n/3)*(Floor(n/3)-1)/2 : n in [0..100]]; // Wesley Ivan Hurt, Apr 05 2015

Magma

[n le 7 select Floor(n/7) else Self(n-1)+2*Self(n-3)-2*Self(n-4)-Self(n-6)+ Self(n-7): n in [1..70]]; // Vincenzo Librandi, Apr 05 2015

Maple

A211534:=n->floor(n/3)*(floor(n/3)-1)/2: seq(A211534(n), n=0..100); # Wesley Ivan Hurt, Apr 05 2015

Mathematica

t[n_] := t[n] = Flatten[Table[-w + 3 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211534 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {0, 0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Apr 05 2015 *)

PARI

concat([0,0,0,0,0,0], Vec(-x^6/((x-1)^3*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, Feb 17 2015

Formula

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).

a(n) = floor(n/3)*( floor(n/3) - 1 )/2. - Luce ETIENNE, Jul 08 2014

G.f.: -x^6 / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Feb 17 2015

a(n) = Sum_{i=0..n-3} i*0^(i mod 3)/3. - Wesley Ivan Hurt, Apr 05 2015