A171621 Numerator of 1/4 - 1/n^2, each fourth term multiplied by 4.

0, 5, 3, 21, 8, 45, 15, 77, 24, 117, 35, 165, 48, 221, 63, 285, 80, 357, 99, 437, 120, 525, 143, 621, 168, 725, 195, 837, 224, 957, 255, 1085, 288, 1221, 323, 1365, 360, 1517, 399, 1677, 440, 1845, 483, 2021, 528

Offset: 2

Author

Paul Curtz, Dec 13 2009

Keywords

• nonn
• easy

Comments

These are the square roots of the fifth column of the array of denominators mentioned in A171522.

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A005563, A010121, A061037, A078371, A171522.

Programs

Magma

[-(-5+3*(-1)^n)*(-4+n^2)/8: n in [0..100]]; // G. C. Greubel, Sep 19 2018

Maple

A061037 := proc(n) 1/4-1/n^2 ; numer(%) ; end proc:
A171621 := proc(n) if n mod 4 = 2 then 4*A061037(n) ; else A061037(n) ; end if; end proc:
seq(A171621(n),n=2..90) ; # R. J. Mathar, Apr 02 2011

Mathematica

Table[-(-5+3*(-1)^n)*(-4+n^2)/8, {n,0,100}] (* G. C. Greubel, Sep 19 2018 *)
LinearRecurrence[{0,3,0,-3,0,1},{0,5,3,21,8,45},50] (* Harvey P. Dale, Nov 01 2019 *)

PARI

concat(0, Vec(x^3*(-5-3*x-6*x^2+x^3+3*x^4)/((x-1)^3*(1+x)^3) + O(x^100))) \\ Colin Barker, Nov 03 2014

Formula

a(n) = A061037(n) * A010121(n+2).

a(2n+2) = A005563(n). a(2n+3) = A078371(n).

G.f.: x^3*(-5-3*x-6*x^2+x^3+3*x^4) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Apr 02 2011

a(n) = -(-5+3*(-1)^n)*(-4+n^2)/8. - Colin Barker, Nov 03 2014

Sum_{n>=3} 1/a(n) = 13/12. - Amiram Eldar, Aug 11 2022