A256833 a(n) = (4*n+3)*(4*n+2).

6, 42, 110, 210, 342, 506, 702, 930, 1190, 1482, 1806, 2162, 2550, 2970, 3422, 3906, 4422, 4970, 5550, 6162, 6806, 7482, 8190, 8930, 9702, 10506, 11342, 12210, 13110, 14042, 15006, 16002, 17030, 18090, 19182, 20306, 21462, 22650, 23870, 25122, 26406

Offset: 0

Author

Bruce Zimov, Apr 10 2015

Keywords

• nonn
• easy

Comments

Since 0 = Sin(Pi) = Sum_{n>=0}(-1)^n*Pi^(2n+1)/(2n+1)!, we can move the negative terms to the other side of the equation to get: Sum_{n>=0} Pi^(4n+1)/(4n+1)! = Sum_{n>=0}Pi^(4n+3)/(4n+3)!.

Now, if we let f(n) = Pi^(4n+1)/(4n+1)!, then the previous equation can be written as Sum_{n>=0}f(n) = Sum_{n>=0}(Pi^2/((4*n+3)*(4*n+2)))*f(n); a(n) is the n-th denominator on the right hand side.

Links

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

Crossrefs

Cf. A000217, A002378, A004767, A016825, A033567.

Programs

Magma

[16*n^2 + 20*n + 6: n in [0..40]]; // Vincenzo Librandi, Apr 12 2015

Mathematica

CoefficientList[Series[(6 + 24 x + 2 x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 12 2015 *)

PARI

vector(50,n,(4*n-1)*(4*n-2)) \\ Derek Orr, Apr 13 2015

Formula

a(n) = 16*n^2 + 20*n + 6.

a(n) = 2*A033567(n+1).

G.f.: (6+24*x+2*x^2)/(1-x)^3. - Vincenzo Librandi, Apr 12 2015

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Vincenzo Librandi, Apr 12 2015

a(n) = A016825(n)*A004767(n). - Tom Edgar, Apr 12 2015

a(n) = A002378(4*n+2) = 2*A000217(4*n+2). - Ivan N. Ianakiev, Apr 17 2015

E.g.f.: 2*exp(x)*(3+18*x+8*x^2). - Wesley Ivan Hurt, Apr 29 2020

From Amiram Eldar, Jan 03 2022: (Start)

Sum_{n>=0} 1/a(n) = Pi/8 - log(2)/4.

Sum_{n>=0} (-1)^n/a(n) = sqrt(2)*log(sqrt(2)+1)/4 - (sqrt(2)-1)*Pi/8. (End)

Extensions

More terms from Vincenzo Librandi, Apr 12 2015