A159864 Difference array of Fibonacci numbers A000045 read by antidiagonals.

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

Offset: 0

Author

Philippe Deléham, Apr 24 2009

Keywords

• easy
• sign
• tabl

Examples

			Triangle begins:
  0;
  1,  1;
  1,  0, -1;
  2,  1,  1,  2;
  3,  1,  0, -1, -3;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A000045, A074331.

Main diagonal gives A039834.

Programs

Maple

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
# second Maple program:
T:= (n, k)-> (<<0|1>, <1|1>>^(n-2*k))[1, 2]:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 27 2022

Mathematica

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[ 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 *)

PARI

{T(n, k) = If(k<0 || k>n, 0, fibonacci(n - 2*k))}; /* Michael Somos, Oct 27 2022 */

Formula

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

(1/2) * Sum_{k=0..n} |T(n,k)| = A074331(n). - Alois P. Heinz, Oct 27 2022

Extensions

Sign of a(65) = -55 corrected by Jean-François Alcover, Apr 14 2015