OFFSET
0,3
COMMENTS
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.
LINKS
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).
FORMULA
EXAMPLE
The coefficients of t^2, t, 1 for t^(-n) begin, for n >= -3:
n t^2 t 1
-----------------
-3 1 1 1
-2 1 0 0
-1 0 1 0
----------------
+0 0 0 1
+1 1 -1 -1
+2 -1 2 0
+3 0 -1 2
+4 2 -2 -3
+5 -3 5 1
+6 1 -4 4
+7 4 -3 -8
+8 -8 12 5
+9 5 -13 7
10 7 -2 -20
...
PROG
(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
a(n) = -(a057597(n+1)-a057597(n)) \\ Felix Fröhlich, Oct 23 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Oct 23 2018
STATUS
approved