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A083239 First order recursion: a(0) = 1; a(n) = phi(n) - a(n-1) = A000010(n) - a(n-1).

%I A083239 #26 Aug 04 2024 20:36:18
%S A083239 1,0,1,1,1,3,-1,7,-3,9,-5,15,-11,23,-17,25,-17,33,-27,45,-37,49,-39,
%T A083239 61,-53,73,-61,79,-67,95,-87,117,-101,121,-105,129,-117,153,-135,159,
%U A083239 -143,183,-171,213,-193,217,-195,241,-225,267,-247,279,-255,307,-289,329,-305,341,-313,371,-355,415,-385,421,-389,437,-417
%N A083239 First order recursion: a(0) = 1; a(n) = phi(n) - a(n-1) = A000010(n) - a(n-1).
%C A083239 Provides interesting decomposition: phi(n) = u+w, where u and w consecutive terms of this sequence. Depends also on initial value.
%H A083239 Amiram Eldar, <a href="/A083239/b083239.txt">Table of n, a(n) for n = 0..10000</a>
%F A083239 a(n) + a(n-1) = A000010(n).
%F A083239 a(n) = (-1)^n * (1 - A068773(n)) for n >= 1. - _Amiram Eldar_, Mar 05 2024
%p A083239 A083239 := proc(n)
%p A083239     option remember ;
%p A083239     if n = 0 then
%p A083239         1 ;
%p A083239     else
%p A083239         numtheory[phi](n)-procname(n-1) ;
%p A083239     end if;
%p A083239 end proc:
%p A083239 seq(A083239(n),n=0..100) ; # _R. J. Mathar_, Jun 20 2021
%t A083239 a[n_] := a[n] = EulerPhi[n] -a[n-1]; a[0] = 1; Table[a[n], {n, 0, 100}]
%o A083239 (Python)
%o A083239 # uses programs from A002088 and A049690
%o A083239 def A083239(n): return A002088(n)-(A049690(n>>1)<<1)-1 if n&1 else 1+(A049690(n>>1)<<1)-A002088(n) # _Chai Wah Wu_, Aug 04 2024
%Y A083239 Cf. A000010, A068773, A083236, A083237, A083238.
%K A083239 sign,easy
%O A083239 0,6
%A A083239 _Labos Elemer_, Apr 23 2003
%E A083239 a(0)=1 prepended by _R. J. Mathar_, Jun 20 2021