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
KEYWORD
nonn,easy,less
AUTHOR
Vincenzo Librandi, Sep 27 2009
EXTENSIONS
Definition simplified by R. J. Mathar, Sep 29 2009
STATUS
approved