A069076 a(n) = (4*n^2 - 1)^3.

27, 3375, 42875, 250047, 970299, 2924207, 7414875, 16581375, 33698267, 63521199, 112678587, 190109375, 307546875, 480048687, 726572699, 1070599167, 1540798875, 2171747375, 3004685307, 4088324799, 5479701947, 7245075375

Offset: 1

Author

Benoit Cloitre, Apr 05 2002

Keywords

• easy
• nonn

References

• Konrad Knopp, Theory and application of infinite series, Dover, p. 269.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000466, A069075.

Programs

Mathematica

(4Range[30]^2-1)^3 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{27,3375,42875,250047,970299,2924207,7414875},30] (* Harvey P. Dale, Jan 20 2012 *)

Formula

Sum_{n>=1} 1/a(n) = (32 - 3*Pi^3)/64.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(1)=27, a(2)=3375, a(3)=42875, a(4)=250047, a(5)=970299, a(6)=2924207, a(7)=7414875. - Harvey P. Dale, Jan 20 2012

G.f: x*(x^6 - 34*x^5 - 3165*x^4 - 19852*x^3 - 19817*x^2 - 3186*x - 27)/(x-1)^7. - Harvey P. Dale, Jan 20 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^3/128 + 3*Pi/32 - 1/2. - Amiram Eldar, Feb 25 2022