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%I #37 Aug 18 2022 10:28:47
%S 0,6,36,66,78,120,210,276,300,378,528,630,666,780,990,1128,1176,1326,
%T 1596,1770,1830,2016,2346,2556,2628,2850,3240,3486,3570,3828,4278,
%U 4560,4656,4950,5460,5778,5886,6216,6786,7140,7260,7626,8256,8646,8778,9180
%N Triangular numbers of the form 6*k.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-5,7,-7,5,-3,1).
%F a(n) = 6 * A154293(n). - _Joerg Arndt_, Aug 18 2022
%F a(n) = A000217(A112652(n+1)-1). - _R. J. Mathar_, Aug 21 2007
%F From _R. J. Mathar_, Nov 18 2009: (Start)
%F 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).
%F G.f.: 6*x*(x^2-x+1)*(x^2+4*x+1)/((1+x^2)^2*(1-x)^3) (6*A154293). (End)
%F From _Amiram Eldar_, Aug 18 2022: (Start)
%F a(n) = A000217(A108752(n)).
%F Sum_{n>=2} 1/a(n) = 2 - (3+4*sqrt(3))*Pi/18. (End)
%p 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);
%t CoefficientList[ Series[ 6x (x^2 -x +1) (x^2 +4x +1)/((x^2 +1)^2*(1 -x)^3), {x, 0, 45}], x] (* or *)
%t LinearRecurrence[{3, -5, 7, -7, 5, -3, 1}, {0, 6, 36, 66, 78, 120, 210}, 46] (* _Robert G. Wilson v_, May 31 2017 *)
%t Select[Accumulate[Range[0, 89]], Divisible[#, 6] &] (* _Alonso del Arte_, May 31 2017 *)
%Y Cf. A000217, A108752, A112652, A154293.
%K nonn,easy
%O 1,2
%A _Amarnath Murthy_, Mar 30 2002
%E More terms from _Sascha Kurz_, Apr 01 2002
%E More terms from _R. J. Mathar_, Aug 21 2007
%E Offset corrected to 1, _Joerg Arndt_, Aug 18 2022
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