A069497 Triangular numbers of the form 6*k.

0, 6, 36, 66, 78, 120, 210, 276, 300, 378, 528, 630, 666, 780, 990, 1128, 1176, 1326, 1596, 1770, 1830, 2016, 2346, 2556, 2628, 2850, 3240, 3486, 3570, 3828, 4278, 4560, 4656, 4950, 5460, 5778, 5886, 6216, 6786, 7140, 7260, 7626, 8256, 8646, 8778, 9180

Offset: 1

Author

Amarnath Murthy, Mar 30 2002

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Crossrefs

Cf. A000217, A108752, A112652, A154293.

Programs

Maple

a[0] := 0:a[1] := 3:a[2] := 8:a[3] := 11:seq((12*(floor(i/4))+a[i mod 4])*(12*(floor(i/4))+a[i mod 4]+1)/2,i=0..100);

Mathematica

CoefficientList[ Series[ 6x (x^2 -x +1) (x^2 +4x +1)/((x^2 +1)^2*(1 -x)^3), {x, 0, 45}], x] (* or *)
LinearRecurrence[{3, -5, 7, -7, 5, -3, 1}, {0, 6, 36, 66, 78, 120, 210}, 46] (* Robert G. Wilson v, May 31 2017 *)
Select[Accumulate[Range[0, 89]], Divisible[#, 6] &] (* Alonso del Arte, May 31 2017 *)

Formula

a(n) = 6 * A154293(n). - Joerg Arndt, Aug 18 2022

a(n) = A000217(A112652(n+1)-1). - R. J. Mathar, Aug 21 2007

From R. J. Mathar, Nov 18 2009: (Start)

a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7).

G.f.: 6*x*(x^2-x+1)*(x^2+4*x+1)/((1+x^2)^2*(1-x)^3) (6*A154293). (End)

From Amiram Eldar, Aug 18 2022: (Start)

a(n) = A000217(A108752(n)).

Sum_{n>=2} 1/a(n) = 2 - (3+4*sqrt(3))*Pi/18. (End)

Extensions

More terms from Sascha Kurz, Apr 01 2002

More terms from R. J. Mathar, Aug 21 2007

Offset corrected to 1, Joerg Arndt, Aug 18 2022