A094585 Triangle T of all positive differences of distinct Fibonacci numbers; also, triangle of all sums of consecutive distinct Fibonacci numbers.

1, 2, 3, 3, 5, 6, 5, 8, 10, 11, 8, 13, 16, 18, 19, 13, 21, 26, 29, 31, 32, 21, 34, 42, 47, 50, 52, 53, 34, 55, 68, 76, 81, 84, 86, 87, 55, 89, 110, 123, 131, 136, 139, 141, 142, 89, 144, 178, 199, 212, 220, 225, 228, 230, 231, 144, 233, 288, 322, 343, 356, 364, 369, 372, 374, 375

Offset: 1

Author

Clark Kimberling, May 13 2004

Keywords

• nonn
• tabl

Comments

Row sums = (1,5,14,34,74,...) = A094584. Alternating row sums = (1,1,4,4,12,12,...) given by F(m+1)-1 if m is even and F(m+2)-1 if m is odd. Central numbers = (1,5,16,47,...) = A094586.

Let p(n,x) = Sum_{k=0..n} F(k+1)*x^(n-k) and q(n,x) = x * q(n-1,x)+1, with q(0,x)=1. Then A094585 is the fission of sequence (p(n,x)) by sequence (q(n,x)); see A193842 for the definition of fission. A094585 is the mirror of A193999. - Clark Kimberling, Aug 11 2011

Examples

			Rows 1 to 5:
  1;
  2,  3;
  3,  5,  6;
  5,  8, 10, 11;
  8, 13, 16, 18, 19;
T(5,4) = F(8) - F(4) = 21 - 3 = 18;
T(5,4) = F(6) + F(5) + F(4) + F(3) = 8 + 5 + 3 + 2 = 18.
		

Links

• Muniru A Asiru, Rows n=1..150 of triangle, flattened

Crossrefs

Cf. A000045, A094584, A094586, A193999.

Programs

GAP

Flat(List([1..11],n->List([1..n],k->Fibonacci(n+3)-Fibonacci(n-k+3)))); # Muniru A Asiru, Apr 28 2019

Mathematica

(* See A193999. *)
Table[Fibonacci[n+3]-Fibonacci[n+3-k],{n,1,20}, {k,1,n}]//TableForm (* Rigoberto Florez, Oct 03 2019 *)

Formula

T(n, k) = F(n+3) - F(n+3-k) = F(n+1) + F(n) + ... + F(n+2-k), for k=1..n; n >= 1.

G.f.: x*y*(x*y+x+1)/((1-y*x)*(x^2+x-1)*(x^2*y^2+x*y-1)). - Vladimir Kruchinin, Jun 20 2025