A017270 - OEIS

%I #31 Nov 30 2024 16:23:22

%S 0,100,400,900,1600,2500,3600,4900,6400,8100,10000,12100,14400,16900,

%T 19600,22500,25600,28900,32400,36100,40000,44100,48400,52900,57600,

%U 62500,67600,72900,78400,84100,90000,96100,102400,108900,115600,122500,129600,136900,144400

%N a(n) = (10*n)^2.

%H Vincenzo Librandi, <a href="/A017270/b017270.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 a(n) = a(n-1) + 200*n - 100, n > 0 ; a(0)=0. - _Miquel Cerda_, Oct 30 2016

%F G.f.: 100*x*(1 + x)/(1 - x)^3. - _Ilya Gutkovskiy_, Oct 30 2016

%F a(n) = 100*A000290(n). - _Michel Marcus_, Oct 30 2016

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

%F Sum_{n>=1} 1/a(n) = Pi^2/600.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/1200.

%F Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/10)/(Pi/10).

%F Product_{n>=1} (1 - 1/a(n)) = sin(Pi/10)/(Pi/10) = 5*(sqrt(5)-1)/(2*Pi). (End)

%F From _Elmo R. Oliveira_, Nov 30 2024: (Start)

%F E.g.f.: 100*x*(1 + x)*exp(x).

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

%F a(n) = A008592(n)^2 = A000290(A008592(n)) = A016850(2*n). (End)

%t LinearRecurrence[{3,-3,1},{0,100,400},40] (* _Harvey P. Dale_, Oct 02 2017 *)

%o (Magma) [(10*n)^2: n in [0..40]]; // _Vincenzo Librandi_, Jul 28 2011

%o (PARI) a(n)=(10*n)^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000290, A008592, A016850.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_