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%I #28 Aug 11 2022 08:54:29
%S 0,5,3,21,8,45,15,77,24,117,35,165,48,221,63,285,80,357,99,437,120,
%T 525,143,621,168,725,195,837,224,957,255,1085,288,1221,323,1365,360,
%U 1517,399,1677,440,1845,483,2021,528
%N Numerator of 1/4 - 1/n^2, each fourth term multiplied by 4.
%C These are the square roots of the fifth column of the array of denominators mentioned in A171522.
%H Colin Barker, <a href="/A171621/b171621.txt">Table of n, a(n) for n = 2..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F a(n) = A061037(n) * A010121(n+2).
%F a(2n+2) = A005563(n). a(2n+3) = A078371(n).
%F 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
%F a(n) = -(-5+3*(-1)^n)*(-4+n^2)/8. - _Colin Barker_, Nov 03 2014
%F Sum_{n>=3} 1/a(n) = 13/12. - _Amiram Eldar_, Aug 11 2022
%p A061037 := proc(n) 1/4-1/n^2 ; numer(%) ; end proc:
%p A171621 := proc(n) if n mod 4 = 2 then 4*A061037(n) ; else A061037(n) ; end if; end proc:
%p seq(A171621(n),n=2..90) ; # _R. J. Mathar_, Apr 02 2011
%t Table[-(-5+3*(-1)^n)*(-4+n^2)/8, {n,0,100}] (* _G. C. Greubel_, Sep 19 2018 *)
%t LinearRecurrence[{0,3,0,-3,0,1},{0,5,3,21,8,45},50] (* _Harvey P. Dale_, Nov 01 2019 *)
%o (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
%o (Magma) [-(-5+3*(-1)^n)*(-4+n^2)/8: n in [0..100]]; // _G. C. Greubel_, Sep 19 2018
%Y Cf. A005563, A010121, A061037, A078371, A171522.
%K nonn,easy
%O 2,2
%A _Paul Curtz_, Dec 13 2009
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