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%I #41 Feb 21 2023 02:11:39
%S 0,11,33,66,110,165,231,308,396,495,605,726,858,1001,1155,1320,1496,
%T 1683,1881,2090,2310,2541,2783,3036,3300,3575,3861,4158,4466,4785,
%U 5115,5456,5808,6171,6545,6930,7326,7733,8151,8580,9020,9471,9933,10406,10890
%N 11 times triangular numbers.
%C Sequence found by reading the line from 0, in the direction 0, 11,... and the same line from 0, in the direction 0, 33,..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Axis perpendicular to A195149 in the same spiral. - _Omar E. Pol_, Sep 18 2011
%C Sum of the numbers from 5n to 6n. - _Wesley Ivan Hurt_, Dec 22 2015
%H G. C. Greubel, <a href="/A152740/b152740.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 11*n*(n+1)/2 = 11*A000217(n).
%F a(n) = a(n-1)+11*n with n>0, a(0)=0. - _Vincenzo Librandi_, Nov 26 2010
%F a(n) = A069125(n+1) - 1. - _Omar E. Pol_, Oct 03 2011
%F From _Philippe Deléham_, Mar 27 2013: (Start)
%F G.f.: 11*x/(1-x)^3.
%F a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) for n>2, a(0)=0, a(1)=11, a(2)=33.
%F a(n) = A218530(11n+10).
%F a(n) = A211013(n)+n = A022269(n)+5n = A022268(n)+6n = A180223(n)+9n = A051865(n)+10n. (End)
%F a(n) = Sum_{i=5n..6n} i. - _Wesley Ivan Hurt_, Dec 22 2015
%F From _Amiram Eldar_, Feb 21 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 2/11.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/11.
%F Product_{n>=1} (1 - 1/a(n)) = -(11/(2*Pi))*cos(sqrt(19/11)*Pi/2).
%F Product_{n>=1} (1 + 1/a(n)) = (11/(2*Pi))*cos(sqrt(3/11)*Pi/2). (End)
%p A152740:=n->11*n*(n+1)/2: seq(A152740(n), n=0..60); # _Wesley Ivan Hurt_, Dec 22 2015
%t Table[11*n*(n - 1)/2, {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)
%t LinearRecurrence[{3, -3, 1}, {0, 11, 33}, 100] (* _G. C. Greubel_, Dec 22 2015 *)
%o (Magma) [11*n*(n+1)/2 : n in [0..60]]; // _Wesley Ivan Hurt_, Dec 22 2015
%o (PARI) my(x='x+O('x^100)); concat(0, Vec(11*x/(1-x)^3)) \\ _Altug Alkan_, Dec 23 2015
%Y Cf. A000217, A022268, A022269, A049598, A051865, A069125, A124080, A180223, A195149, A195313, A211013, A218530.
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Dec 12 2008
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