A001535 a(n) = (10n+1)*(10n+9).

9, 209, 609, 1209, 2009, 3009, 4209, 5609, 7209, 9009, 11009, 13209, 15609, 18209, 21009, 24009, 27209, 30609, 34209, 38009, 42009, 46209, 50609, 55209, 60009, 65009, 70209, 75609, 81209, 87009, 93009, 99209, 105609, 112209, 119009, 126009, 133209, 140609

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001622, A017281, A017377.

Programs

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 *)

PARI

a(n)=(10*n+1)*(10*n+9) \\ Charles R Greathouse IV, Jun 16 2017

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)

a(n) = A017281(n)*A017377(n).

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)