A154292 - OEIS

%I #23 Mar 18 2022 05:20:53

%S 11,13,46,50,105,111,188,196,295,305,426,438,581,595,760,776,963,981,

%T 1190,1210,1441,1463,1716,1740,2015,2041,2338,2366,2685,2715,3056,

%U 3088,3451,3485,3870,3906,4313,4351,4780,4820,5271,5313,5786,5830,6325,6371

%N Integers of the form m*(6*m -+ 1)/2.

%H Colin Barker, <a href="/A154292/b154292.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F From _Colin Barker_, Feb 26 2016: (Start)

%F a(n) = (12*n^2 - 10*(-1)^n*n + 12*n - 5*(-1)^n + 5)/4.

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.

%F G.f.: x*(11 + 2*x + 11*x^2) / ((1-x)^3*(1+x)^2). (End)

%F E.g.f.: (1/4)*(-5 + 10*x + (5 + 24*x + 12*x^2)*exp(2*x))*exp(-x). - _G. C. Greubel_, Sep 10 2016

%F From _Amiram Eldar_, Mar 18 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 131/11 - (2+sqrt(3))*Pi.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 133/11 - 3*log(12) - 2*sqrt(3)*log(2+sqrt(3)). (End)

%t Flatten[Table[{n (6n-1)/2,n (6n+1)/2},{n,2,50,2}]] (* _Harvey P. Dale_, Jan 19 2013 *)

%o (PARI) Vec(x*(11+2*x+11*x^2)/((1-x)^3*(1+x)^2) + O(x^60)) \\ _Colin Barker_, Feb 26 2016

%o (Magma) &cat[[n*(6*n-1) div 2, n*(6*n+1) div 2]: n in [2..60 by 2]]; // _Vincenzo Librandi_, Sep 10 2016

%Y Cf. A001318, A074378, A057569, A057570.

%K nonn,easy

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Jan 06 2009