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A002121 a(0) = 1, a(1) = 0, a(2) = -1; for n >= 3, a(n) = - a(n-2) + Sum_{ primes p with 3 <= p <= n} a(n-p).

%I A002121 M0023 N0005 #26 Oct 14 2023 22:35:18
%S A002121 1,0,-1,1,1,-1,0,2,0,-2,2,4,-3,-2,8,1,-8,8,12,-11,-4,25,4,-24,21,40,
%T A002121 -31,-16,82,14,-81,71,131,-99,-48,258,46,-249,223,422,-303,-162,825,
%U A002121 169,-791,714,1360,-955,-503,2641,573,-2479,2263,4365,-2941,-1592,8436,1978,-7830,7212,14083,-9133,-4992,26970,6688,-24590
%N A002121 a(0) = 1, a(1) = 0, a(2) = -1; for n >= 3, a(n) = - a(n-2) + Sum_{ primes p with 3 <= p <= n} a(n-p).
%C A002121 Arises in studying the Goldbach conjecture.
%C A002121 The last negative term appears to be a(303). - _T. D. Noe_, Dec 05 2006
%D A002121 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002121 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002121 T. D. Noe, <a href="/A002121/b002121.txt">Table of n, a(n) for n = 0..1000</a>
%H A002121 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-23.1.290">Properties of prime numbers deduced from the calculus of symmetric functions</a>, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380. [The sequence g_n]
%H A002121 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F A002121 G.f.: 1/(1+Sum_{k>0} (-x)^prime(k)). - _Vladeta Jovovic_, Mar 29 2003
%t A002121 CoefficientList[Series[1/(1+Sum[(-x)^Prime[k],{k,20}]),{x,0,70}],x] (* _Harvey P. Dale_, Aug 26 2020 *)
%o A002121 (Haskell)
%o A002121 import Data.List (genericIndex)
%o A002121 a002121 n = genericIndex a002121_list n
%o A002121 a002121_list = 1 : 0 : -1 : f 0 (-1) 3 where
%o A002121    f v w x = y : f w y (x + 1) where
%o A002121      y = sum (map (a002121 . (x -)) $ takeWhile (<= x) a065091_list) - v
%o A002121 -- _Reinhard Zumkeller_, Mar 21 2014
%Y A002121 Cf. A002100-A002125.
%Y A002121 Cf. A065091.
%K A002121 sign,easy,look
%O A002121 0,8
%A A002121 _N. J. A. Sloane_
%E A002121 More terms from _Vladeta Jovovic_, Mar 29 2003
%E A002121 Entry revised by _N. J. A. Sloane_, Dec 04 2006