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

Offset

1,2

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]

Links

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

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000045, A094584, A094586, A193999.

Sequence in context: A051599 A207292 A064464 * A183322 A295918 A296834

Adjacent sequences: A094582 A094583 A094584 * A094586 A094587 A094588

Keyword

nonn,tabl

Author

Clark Kimberling, May 13 2004

Status

approved