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Difference array of Fibonacci numbers A000045 read by antidiagonals.
1

%I #23 Oct 27 2022 19:43:07

%S 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 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 1,0,-1,-3,-8,-21,-55,89,34,13,5,2,1,1,2,5,13,34,89

%N Difference array of Fibonacci numbers A000045 read by antidiagonals.

%H Alois P. Heinz, <a href="/A159864/b159864.txt">Rows n = 0..200, flattened</a>

%F 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 (1/2) * Sum_{k=0..n} |T(n,k)| = A074331(n). - _Alois P. Heinz_, Oct 27 2022

%e Triangle begins:

%e 0;

%e 1, 1;

%e 1, 0, -1;

%e 2, 1, 1, 2;

%e 3, 1, 0, -1, -3;

%e ...

%p 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 # second Maple program:

%p T:= (n, k)-> (<<0|1>, <1|1>>^(n-2*k))[1, 2]:

%p seq(seq(T(n, k), k=0..n), n=0..12); # _Alois P. Heinz_, Oct 27 2022

%t 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 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 (PARI) {T(n, k) = If(k<0 || k>n, 0, fibonacci(n - 2*k))}; /* _Michael Somos_, Oct 27 2022 */

%Y Cf. A000045, A074331.

%Y Main diagonal gives A039834.

%K easy,sign,tabl

%O 0,7

%A _Philippe Deléham_, Apr 24 2009

%E Sign of a(65) = -55 corrected by _Jean-François Alcover_, Apr 14 2015