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%I #27 Sep 08 2022 08:45:48
%S 65,260,585,1040,1625,2340,3185,4160,5265,6500,7865,9360,10985,12740,
%T 14625,16640,18785,21060,23465,26000,28665,31460,34385,37440,40625,
%U 43940,47385,50960,54665,58500,62465,66560,70785,75140,79625,84240
%N a(n) = 65*n^2.
%C Alternative definition: Numbers m>0 such that 65*m is square.
%C Also the sum of the areas of the two squares that equals the area of a rectangle with whole number sides using the formula x^2 + y^2 = (x+y+sqrt(2*x*y))(x+y-sqrt(2*x*y))where the substitution y=8*x obtains the whole number sides of the rectangle. Thus x^2 + (8x)^2 =(13*x)(5*x) or 65*x^2 = 13*x*(5*x)
%C x Squares Sum Rectangle (l,w) Area
%C 1 1,64 65 13,5 65
%C 2 4,256 260 26,10 260 -_Larry J Zimmermann_, Feb 28 2013
%C From _G. C. Greubel_, Apr 08 2016: (Start)
%C a(n) (mod 10) = 5*A000035(n).
%C a(n) (mod 10) = 5*A059841(n-1). (End)
%H Vincenzo Librandi, <a href="/A165798/b165798.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _R. J. Mathar_, Sep 28 2009: (Start)
%F a(n) = 65*A000290(n).
%F G.f.: 65*x*(1+x)/(1-x)^3. (End)
%F From _G. C. Greubel_, Apr 08 2016: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F E.g.f.: 65*x*(1+x)*exp(x). (End)
%t Table[65*n^2,{n,1,40}] (* _Vincenzo Librandi_, Mar 14 2012 *)
%t LinearRecurrence[{3,-3,1},{65,260,585},40] (* _Harvey P. Dale_, Jan 12 2020 *)
%o (Magma) [65*n^2: n in [1..40]]; // _Vincenzo Librandi_, Mar 14 2012
%o (PARI) a(n)=65*n^2 \\ _Charles R Greathouse IV_, Jun 17 2017
%K nonn,easy,less
%O 1,1
%A _Vincenzo Librandi_, Sep 27 2009
%E Definition simplified by _R. J. Mathar_, Sep 29 2009