OFFSET
0,1
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=9, a(1)=209, a(2)=609. - Harvey P. Dale, Oct 15 2014
G.f.: (9 + 182*x + 9*x^2)/(1 - x)^3. - Vincenzo Librandi, Dec 17 2014
E.g.f.: (100*x^2 + 200*x + 9)*exp(x). - Robert Israel, Dec 17 2014
From Amiram Eldar, Feb 20 2023: (Start)
Sum_{n>=0} 1/a(n) = sqrt(5+2*sqrt(5))*Pi/80.
Sum_{n>=0} (-1)^n/a(n) = (sqrt(10+2*sqrt(5)) * log(cot(Pi/20)) + sqrt(10-2*sqrt(5)) * log(cot(3*Pi/20)))/80.
Product_{n>=0} (1 - 1/a(n)) = 2*phi*cos(sqrt(17)*Pi/10), where phi is the golden ratio (A001622).
Product_{n>=0} (1 + 1/a(n)) = 2*phi*cos(sqrt(15)*Pi/10). (End)
MAPLE
seq((10*n+1)*(10*n+9), n = 0 .. 100); # Robert Israel, Dec 17 2014
MATHEMATICA
Times@@@Table[10n+{1, 9}, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {9, 209, 609}, 40] (* Harvey P. Dale, Oct 15 2014 *)
CoefficientList[Series[(9 + 182 x + 9 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 17 2014 *)
PROG
(PARI) a(n)=(10*n+1)*(10*n+9) \\ Charles R Greathouse IV, Jun 16 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved