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A165798 a(n) = 65*n^2. 1

%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

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Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)