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A227431 Fibonacci differences triangle, T(n,k), k<=n, where column k holds the k-th difference of A000045, read by rows.

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