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A159864 Difference array of Fibonacci numbers A000045 read by antidiagonals.

%I A159864 #23 Oct 27 2022 19:43:07
%S A159864 0,1,1,1,0,-1,2,1,1,2,3,1,0,-1,-3,5,2,1,1,2,5,8,3,1,0,-1,-3,-8,13,5,2,
%T A159864 1,1,2,5,13,21,8,3,1,0,-1,-3,-8,-21,34,13,5,2,1,1,2,5,13,34,55,21,8,3,
%U A159864 1,0,-1,-3,-8,-21,-55,89,34,13,5,2,1,1,2,5,13,34,89
%N A159864 Difference array of Fibonacci numbers A000045 read by antidiagonals.
%H A159864 Alois P. Heinz, <a href="/A159864/b159864.txt">Rows n = 0..200, flattened</a>
%F A159864 Conjecture: row sums are Sum_{k=0..n} T(2n,k)=0. Sum_{k=0..n} T(2n+1,k) = A025169(n). - _R. J. Mathar_, May 29 2009
%F A159864 (1/2) * Sum_{k=0..n} |T(n,k)| = A074331(n). - _Alois P. Heinz_, Oct 27 2022
%e A159864 Triangle begins:
%e A159864   0;
%e A159864   1,  1;
%e A159864   1,  0, -1;
%e A159864   2,  1,  1,  2;
%e A159864   3,  1,  0, -1, -3;
%e A159864   ...
%p A159864 A159864Q := proc(n,k) option remember; if n = 0 then combinat[fibonacci](k) ; else procname(n-1,k+1) -procname(n-1,k) ; fi; end: A159864 := proc(n,k) A159864Q(k,n-k) ; end: for n from 0 to 5 do for k from 0 to n do printf("%d,",A159864(n,k)) ; od: od: # _R. J. Mathar_, May 29 2009
%p A159864 # second Maple program:
%p A159864 T:= (n, k)-> (<<0|1>, <1|1>>^(n-2*k))[1, 2]:
%p A159864 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Oct 27 2022
%t A159864 nmax = 10; f = Table[Fibonacci[n], {n, 0, nmax}]; t = Table[Differences[f, n], {n, 0, nmax}]; Table[t[[n-k+1, k+1]], {n, 0, nmax}, {k, n, 0, -1}]  // Flatten (* _Jean-François Alcover_, Apr 14 2015 *)
%t A159864 T[ n_, k_] := If[ k<0 || k>n, 0, Fibonacci[n - 2*k]]; Join@@Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* _Michael Somos_, Oct 27 2022 *)
%o A159864 (PARI) {T(n, k) = If(k<0 || k>n, 0, fibonacci(n - 2*k))}; /* _Michael Somos_, Oct 27 2022 */
%Y A159864 Cf. A000045, A074331.
%Y A159864 Main diagonal gives A039834.
%K A159864 easy,sign,tabl
%O A159864 0,7
%A A159864 _Philippe Deléham_, Apr 24 2009
%E A159864 Sign of a(65) = -55 corrected by _Jean-François Alcover_, Apr 14 2015