A133766 a(n) = (4*n+1)*(4*n+3)*(4*n+5).

15, 315, 1287, 3315, 6783, 12075, 19575, 29667, 42735, 59163, 79335, 103635, 132447, 166155, 205143, 249795, 300495, 357627, 421575, 492723, 571455, 658155, 753207, 856995, 969903, 1092315, 1224615, 1367187, 1520415, 1684683, 1860375, 2047875, 2247567, 2459835

Offset: 0

Author

Miklos Kristof, Jan 02 2008

Keywords

• nonn
• easy

References

• L. B. W. Jolley, Summation of Series, Dover, 1961.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A001539, A154633.

Programs

Maple

seq((4*n+1)*(4*n+3)*(4*n+5),n=0..40);

Mathematica

Table[c=4n;(c+1)(c+3)(c+5),{n,0,30}] (* or *) LinearRecurrence[{4,-6,4,-1},{15,315,1287,3315},30] (* Harvey P. Dale, May 06 2012 *)

PARI

a(n)=(4*n+1)*(4*n+3)*(4*n+5) \\ Charles R Greathouse IV, Oct 16 2015

Formula

G.f.: 3*(5 + 85*x + 39*x^2 - x^3)/(1-x)^4 .

E.g.f: (15 + 300*x + 336*x^2 + 64*x^3)*exp(x) .

Sum_{n>=0} 4/a(n) = (Pi-2)/4. [Jolley, eq. 238]

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Harvey P. Dale, May 06 2012

Sum_{n>=0} (-1)^n/a(n) = 1/8 + (log(2*sqrt(2)+3) - Pi)/(16*sqrt(2)). - Amiram Eldar, Feb 27 2022