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

65, 260, 585, 1040, 1625, 2340, 3185, 4160, 5265, 6500, 7865, 9360, 10985, 12740, 14625, 16640, 18785, 21060, 23465, 26000, 28665, 31460, 34385, 37440, 40625, 43940, 47385, 50960, 54665, 58500, 62465, 66560, 70785, 75140, 79625, 84240

Offset

1,1

Comments

Alternative definition: Numbers m>0 such that 65*m is square.

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)

x Squares Sum Rectangle (l,w) Area

1 1,64 65 13,5 65

2 4,256 260 26,10 260 -Larry J Zimmermann, Feb 28 2013

From G. C. Greubel, Apr 08 2016: (Start)

a(n) (mod 10) = 5*A000035(n).

a(n) (mod 10) = 5*A059841(n-1). (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

From R. J. Mathar, Sep 28 2009: (Start)

a(n) = 65*A000290(n).

G.f.: 65*x*(1+x)/(1-x)^3. (End)

From G. C. Greubel, Apr 08 2016: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

E.g.f.: 65*x*(1+x)*exp(x). (End)

Mathematica

Table[65*n^2, {n, 1, 40}] (* Vincenzo Librandi, Mar 14 2012 *)
LinearRecurrence[{3, -3, 1}, {65, 260, 585}, 40] (* Harvey P. Dale, Jan 12 2020 *)

Prog

(Magma) [65*n^2: n in [1..40]]; // Vincenzo Librandi, Mar 14 2012
(PARI) a(n)=65*n^2 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Sequence in context: A031694 A152023 A369498 * A158693 A365874 A319617

Adjacent sequences: A165795 A165796 A165797 * A165799 A165800 A165801

Keyword

nonn,easy,less

Author

Vincenzo Librandi, Sep 27 2009

Extensions

Definition simplified by R. J. Mathar, Sep 29 2009

Status

approved