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%I #28 Jul 01 2024 02:09:42
%S 0,1,-2,-3,4,5,-6,-7,8,9,-10,-11,12,13,-14,-15,16,17,-18,-19,20,21,
%T -22,-23,24,25,-26,-27,28,29,-30,-31,32,33,-34,-35,36,37,-38,-39,40,
%U 41,-42,-43,44,45,-46,-47,48,49,-50,-51,52,53,-54,-55,56,57,-58,-59
%N a(n) = (-1)^floor(n/2)*n.
%C For all odd numbers n (A005408) and all whole numbers z (A001057) K(z/n) = K(a(n)/z), where K(z/n) denotes the Kronecker symbol (A372728). This fact is equivalent to the law of quadratic reciprocity and its first and second supplement.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,-2,0,-1).
%F Sum_{n>=1} 1/a(n) = Pi/4 - log(2)/2 = A196521.
%F a(n) = [x^n] -x*(x^2 + 2*x - 1)/(x^2 + 1)^2.
%F a(n) = n! * [x^n] x*(cos(x) - sin(x)). - _Stefano Spezia_, Jun 30 2024
%F a(n) = n*A057077(n). - _Michel Marcus_, Jul 01 2024
%p a := n -> (-1)^iquo(n, 2)*n: seq(a(n), n = 0..59);
%t Array[(-1)^Floor[#/2]*# &, 60, 0] (* _Michael De Vlieger_, Jun 30 2024 *)
%o (Python)
%o def A374157(n): return (-1)**(n // 2)*n
%o (Python)
%o def A374157(n): return -n if n&2 else n # _Chai Wah Wu_, Jun 30 2024
%o (PARI) a(n) = (-1)^(n\2) * n; \\ _Amiram Eldar_, Jun 30 2024
%Y Cf. A005408, A001057, A057077, A196521, A372728.
%Y Cf. A001477, A181983.
%K sign,easy
%O 0,3
%A _Peter Luschny_, Jun 30 2024
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