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A319200 a(n) = -(A(n) - A(n-1)) where A(n) = A057597(n+1), for n >= 0.

%I A319200 #20 Jul 20 2025 12:49:55
%S A319200 0,-1,2,-1,-2,5,-4,-3,12,-13,-2,27,-38,9,56,-103,56,103,-262,215,150,
%T A319200 -627,692,85,-1404,2011,-522,-2893,5426,-3055,-5264,13745,-11536,
%U A319200 -7473,32754,-36817,-3410,72981,-106388,29997,149372,-285757,166382,268747,-720886,618521,371112,-1710519,1957928,123703,-3792150
%N A319200 a(n) = -(A(n) - A(n-1)) where A(n) = A057597(n+1), for n >= 0.
%C A319200 This sequence appears in the reduction formula for negative powers of the tribonacci constant t = A058265: t^(-n) = A(n)*t^2 + a(n)*t + A(n+1)*1, with A(n) = A057597(n+1), for n >= 0. This follows from t^3 = t^2 +  t + 1, or 1/t = t^2 - t - 1 = A192918, leading to the recurrence: A(n) = -A(n) - A(n-1) + A(n-2), with inputs A(-3) = 1, A(-2) = 1 and A(-1) = 0 and  a(n) = -(A(n) - A(n-1)). See the example below.
%H A319200 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,1).
%F A319200 a(n) = -(A057597(n+1) - A057597(n)), for n >= 0.
%F A319200 Recurrence  a(n) = -a(n-1) - a(n-2) + a(n-3), for n >=0, with a(-3) = 1, a(-2) = 0 and a(-1) = 1.
%F A319200 G.f.: (1 + 1/x)/(1 + x + x^2 - x^3).
%e A319200 The coefficients of t^2, t, 1 for t^(-n) begin, for n >= -3:
%e A319200 n     t^2  t   1
%e A319200 -----------------
%e A319200 -3     1   1   1
%e A319200 -2     1   0   0
%e A319200 -1     0   1   0
%e A319200 ----------------
%e A319200 +0     0   0   1
%e A319200 +1     1  -1  -1
%e A319200 +2    -1   2   0
%e A319200 +3     0  -1   2
%e A319200 +4     2  -2  -3
%e A319200 +5    -3   5   1
%e A319200 +6     1  -4   4
%e A319200 +7     4  -3  -8
%e A319200 +8    -8  12   5
%e A319200 +9     5 -13   7
%e A319200 10     7  -2 -20
%e A319200 ...
%t A319200 LinearRecurrence[{-1,-1,1},{0,-1,2},60] (* _Harvey P. Dale_, Jul 20 2025 *)
%o A319200 (PARI) a057597(n) = polcoeff( if( n<0, x / ( 1 - x - x^2 - x^3), x^2 / ( 1 + x + x^2 - x^3) ) + x*O(x^abs(n)), abs(n)) \\ after _Michael Somos_ in A057597
%o A319200 a(n) = -(a057597(n+1)-a057597(n)) \\ _Felix Fröhlich_, Oct 23 2018
%Y A319200 Cf. A057597, A058265, A078016(n+1) (different signs), A192918.
%K A319200 sign,easy
%O A319200 0,3
%A A319200 _Wolfdieter Lang_, Oct 23 2018