login
a(n) = (12*n+1)*(12*n+11).
4

%I #37 Oct 25 2024 17:05:36

%S 11,299,875,1739,2891,4331,6059,8075,10379,12971,15851,19019,22475,

%T 26219,30251,34571,39179,44075,49259,54731,60491,66539,72875,79499,

%U 86411,93611,101099,108875,116939,125291,133931,142859,152075,161579,171371,181451,191819

%N a(n) = (12*n+1)*(12*n+11).

%H T. D. Noe, <a href="/A001538/b001538.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = (9*A001533(n) - 19)/4.

%F a(n) = 288*n + a(n-1) with a(0)=11. - _Vincenzo Librandi_, Nov 12 2010

%F G.f.: -(11 + 266*x + 11*x^2)/(x-1)^3. - _R. J. Mathar_, Jun 30 2020

%F From _Amiram Eldar_, Feb 20 2023: (Start)

%F a(n) = A017533(n)*A017653(n).

%F Sum_{n>=0} 1/a(n) = (sqrt(3)+2)*Pi/120.

%F Sum_{n>=0} (-1)^n/a(n) = (4*log(sqrt(2)+1) + sqrt(3)*log(5+2*sqrt(6)))/(60*sqrt(2)).

%F Product_{n>=0} (1 - 1/a(n)) = (2*sqrt(2)/(sqrt(3)-1))*cos(sqrt(13/2)*Pi/6).

%F Product_{n>=0} (1 + 1/a(n)) = 2*sqrt(2+sqrt(3))*cos(Pi/sqrt(6)). (End)

%F From _Elmo R. Oliveira_, Oct 25 2024: (Start)

%F E.g.f.: exp(x)*(11 + 144*x*(2 + x)).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

%t Table[(12n+1)(12n+11),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{11,299,875},40] (* _Harvey P. Dale_, Jul 22 2024 *)

%o (PARI) a(n)=(12*n+1)*(12*n+11) \\ _Charles R Greathouse IV_, Jun 16 2017

%Y Cf. A001533, A017533, A017653.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_