A017066 - OEIS

%I #49 Dec 06 2024 15:35:25

%S 0,64,256,576,1024,1600,2304,3136,4096,5184,6400,7744,9216,10816,

%T 12544,14400,16384,18496,20736,23104,25600,28224,30976,33856,36864,

%U 40000,43264,46656,50176,53824,57600,61504,65536,69696,73984,78400,82944,87616,92416,97344,102400

%N a(n) = (8*n)^2.

%H Vincenzo Librandi, <a href="/A017066/b017066.txt">Table of n, a(n) for n = 0..10000</a>

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

%F G.f.: -64*x*(1+x)/(x-1)^3. - _R. J. Mathar_, Jul 14 2016

%F a(n) = A000290(8*n) = A008590(n)^2 = A000290(A008590(n)).

%F From _Amiram Eldar_, Jan 25 2021: (Start)

%F Sum_{n>=1} 1/a(n) = Pi^2/384.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/768.

%F Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/8)/(Pi/8).

%F Product_{n>=1} (1 - 1/a(n)) = sin(Pi/8)/(Pi/8) = 4*sqrt(2-sqrt(2))/Pi. (End)

%F From _Elmo R. Oliveira_, Dec 06 2024: (Start)

%F E.g.f.: 64*exp(x)*x*(1 + x).

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

%F a(n) = n*A152691(n) = 2*A244082(n) = A016802(2*n). (End)

%t LinearRecurrence[{3, -3, 1}, {0, 64, 256}, 50] (* _Vincenzo Librandi_, Feb 10 2012 *)

%o (Magma) [(8*n)^2: n in [0..35]]; // _Vincenzo Librandi_, Jul 10 2011

%o (PARI) a(n)=(8*n)^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000290, A008590, A016802, A152691, A244082.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_