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A094585 Triangle T of all positive differences of distinct Fibonacci numbers; also, triangle of all sums of consecutive distinct Fibonacci numbers. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 28 00:30 EDT 2022. Contains 354110 sequences. (Running on oeis4.)