A177065 a(n) = (8*n+3)*(8*n+5).

15, 143, 399, 783, 1295, 1935, 2703, 3599, 4623, 5775, 7055, 8463, 9999, 11663, 13455, 15375, 17423, 19599, 21903, 24335, 26895, 29583, 32399, 35343, 38415, 41615, 44943, 48399, 51983, 55695, 59535, 63503, 67599, 71823, 76175, 80655, 85263, 89999, 94863, 99855

Offset: 0

Author

Vincenzo Librandi, May 31 2010

Keywords

• nonn
• easy

Comments

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 64*A002061(n+1) - 49. - Bruno Berselli, Aug 24 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002061, A004770, A016754, A017101, A125169, A177059.

Programs

Magma

[(8*n+3)*(8*n+5): n in [0..50]]; // Vincenzo Librandi, Apr 08 2013

Maple

A177065:=n->(8*n+3)*(8*n+5): seq(A177065(n), n=0..100); # Wesley Ivan Hurt, Apr 24 2017

Mathematica

Table[(8n+3)(8n+5),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{15,143,399},40] (* Harvey P. Dale, Mar 13 2013 *)
CoefficientList[Series[(15 + 98 x + 15 x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 08 2013 *)

PARI

a(n)=(8*n+3)*(8*n+5) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 128*n + a(n-1) with n > 0, a(0)=15.

a(n) = A125169(A016754(n) - 1). - Reinhard Zumkeller, Jul 05 2010

a(0)=15, a(1)=143, a(2)=399, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 13 2013

G.f.: (15+98*x+15*x^2)/(1-x)^3. - Vincenzo Librandi, Apr 08 2013

From Amiram Eldar, Feb 19 2023: (Start)

a(n) = A017101(n)*A004770(n).

Sum_{n>=0} 1/a(n) = (sqrt(2)-1)*Pi/16.

Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(tan(3*Pi/16)) + sin(Pi/8) * log(cot(Pi/16)))/4.

Product_{n>=0} (1 - 1/a(n)) = sec(Pi/8)*cos(Pi/(4*sqrt(2))).

Product_{n>=0} (1 + 1/a(n)) = sec(Pi/8). (End)

E.g.f.: exp(x)*(15 + 64*x*(2 + x)). - Elmo R. Oliveira, Oct 25 2024

Extensions

Edited by N. J. A. Sloane, Jun 22 2010