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%I #34 Jul 30 2016 19:53:51
%S 1,1,0,2,1,1,3,1,0,-1,5,2,1,1,2,8,3,1,0,-1,-3,13,5,2,1,1,2,5,21,8,3,1,
%T 0,-1,-3,-8,34,13,5,2,1,1,2,5,13,55,21,8,3,1,0,-1,-3,-8,-21,89,34,13,
%U 5,2,1,1,2,5,13,34,144,55,21,8,3,1,0,-1,-3,-8,-21
%N Fibonacci differences triangle, T(n,k), k<=n, where column k holds the k-th difference of A000045, read by rows.
%C Consecutive columns (i.e., k = 1, 2, 3, ...) shift the Fibonacci sequence down by 2 indices.
%C Diagonal (n = k) produces Fibonacci numbers at increasingly negative indices for n = k > 2. See A039834.
%C Row sums equal A005013(n), which equals Fibonacci A000045(n), if n is even, and equals Lucas numbers A000204(n) if n is odd.
%C (Rows that sum to Lucas numbers have all positive values.)
%H T. D. Noe, <a href="/A227431/b227431.txt">Rows n = 1..100 of triangle, flattened</a>
%F T(n,1) = F(n) for n > 0, where F(n) = A000045(n), T(n,k) = T(n,k-1) - T(n-1,k-1).
%e 1
%e 1 0
%e 2 1 1
%e 3 1 0 -1
%e 5 2 1 1 2
%e 8 3 1 0 -1 -3
%e 13 5 2 1 1 2 5
%e 21 8 3 1 0 -1 -3 -8
%e 34 13 5 2 1 1 2 5 13
%e 55 21 8 3 1 0 -1 -3 -8 -21
%e 89 34 13 5 2 1 1 2 5 13 34
%t Flatten[Table[Fibonacci[Range[n, -n + 1, -2]], {n, 15}]] (* _T. D. Noe_, Jul 26 2013 *)
%o (Haskell)
%o a227431 n k = a227431_tabl !! (n-1) !! (k-1)
%o a227431_row n = a227431_tabl !! (n-1)
%o a227431_tabl = h [] 0 1 where
%o h row u v = row' : h row' v (u + v) where row' = scanl (-) v row
%o -- _Reinhard Zumkeller_, Jul 28 2013
%o (PARI) T(n,k)=fibonacci(n-2*k+2) \\ _Charles R Greathouse IV_, Jul 30 2016
%Y Cf. A000045, A039834, A005013.
%K sign,easy,nice,tabl
%O 1,4
%A _Richard R. Forberg_, Jul 11 2013
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