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A119972 a(n) = n * A034947(n).

%I A119972 #25 Jan 19 2023 15:32:06
%S A119972 1,2,-3,4,5,-6,-7,8,9,10,-11,-12,13,-14,-15,16,17,18,-19,20,21,-22,
%T A119972 -23,-24,25,26,-27,-28,29,-30,-31,32,33,34,-35,36,37,-38,-39,40,41,42,
%U A119972 -43,-44,45,-46,-47,-48,49,50,-51,52,53,-54,-55,-56,57,58,-59,-60,61,-62,-63,64,65,66,-67,68,69,-70,-71,72,73,74,-75
%N A119972 a(n) = n * A034947(n).
%C A119972 Previous name was: Flag n when the first difference of the decimal encoding of the Gray code is negative. (With "flag" meaning negate n when the difference is negative.)
%C A119972 Merge A091072 with minus A091067 maintaining increasing absolute value.
%F A119972 a(n) = n*Kronecker(-1, n) = n * A034947(n). - _Andrew Howroyd_, Aug 06 2018
%e A119972 A003188 begins 0  1  3  2  6  7  5  4 12  13  15  14  10  11  9 ... so
%e A119972 A055975 begins   1  2 -1  4  1 -2 -1  8  1   2  -1  -4   1  -2  ...
%e A119972 Sequence         1, 2,-3, 4, 5,-6,-7, 8, 9, 10,-11,-12, 13,-14, ...
%e A119972 Negative terms are at positions 3,6,7,11,12,14,..., = A091067.
%e A119972 Positive terms are the complement, which is A091072.
%p A119972 isA091067 := proc(n) option remember ; if n mod 4 = 3 then RETURN(true) ; else if n mod 2 = 0 then if isA091067(n/2) then RETURN(true) ; fi ; fi ; RETURN(false) ; fi ; end: A119972 := proc(n) if isA091067(n) then -n ; else n ; fi ; end: for n from 1 to 180 do printf("%d, ",A119972(n)) ; od ; # _R. J. Mathar_, May 14 2007
%p A119972 # second Maple program:
%p A119972 a:= n-> numtheory[jacobi](-1, n)*n:
%p A119972 seq(a(n), n=1..75);  # _Alois P. Heinz_, Jan 19 2023
%t A119972 a[n_] := n KroneckerSymbol[-1, n];
%t A119972 Array[a, 75] (* _Jean-François Alcover_, Apr 09 2020 *)
%o A119972 (PARI) a(n) = n*kronecker(-1, n); \\ _Andrew Howroyd_, Aug 06 2018
%Y A119972 Cf. A034947, A003188, A055975, A091067, A091072.
%K A119972 easy,sign,mult
%O A119972 1,2
%A A119972 _Alford Arnold_, Jun 01 2006
%E A119972 More terms from _R. J. Mathar_, May 14 2007
%E A119972 Keyword:mult added by _Andrew Howroyd_, Aug 06 2018
%E A119972 New name using existing formula from _Joerg Arndt_, Jan 19 2023