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A069497
Triangular numbers of the form 6*k.
3
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
1,2
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)
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amarnath Murthy, Mar 30 2002
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
STATUS
approved