A104762 Triangle read by rows: row n contains first n nonzero Fibonacci numbers in decreasing order.

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 3, 2, 1, 1, 8, 5, 3, 2, 1, 1, 13, 8, 5, 3, 2, 1, 1, 21, 13, 8, 5, 3, 2, 1, 1, 34, 21, 13, 8, 5, 3, 2, 1, 1, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 233, 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1

Offset: 1

Author

Gary W. Adamson, Mar 23 2005, Mar 05 2007

Keywords

• nonn
• easy
• tabl

Comments

Sum of n-th row = F(n+2) - 1; sequence A000071 starting (1, 2, 4, 7, 12, 20, ...).

Riordan array (1/(1-x-x^2),x). - Philippe Deléham, Apr 23 2009 [with offset 0]

Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A104762 is the reverse reluctant sequence of Fibonacci numbers (A000045), except 0. - Boris Putievskiy, Dec 13 2012

Examples

			First few rows of the triangle:
  1;
  1, 1;
  2, 1, 1;
  3, 2, 1, 1;
  5, 3, 2, 1, 1;
  8, 5, 3, 2, 1, 1;
  ...
From _Philippe Deléham_, Oct 07 2014: (Start)
Production matrix begins:
  1, 1;
  1, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  ... (End)
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A000045, A000071, A271355 (analogous Lucas triangle).

Companion triangle A104763, Fibonacci sequence in each row starting from the left. A121461.

Programs

Maple

seq(seq(combinat:-fibonacci(n-i),i=0..n-1),n=1..20); # Robert Israel, May 01 2016

Mathematica

r = N[(1 + Sqrt[5])/2, 100]; s = N[(1 - Sqrt[5])/2, 100];
t = Table[Abs[Round[(r^n)*(s^k)/Sqrt[5]]], {n, 2, 15}, {k, 1, n - 1}]
Flatten[t]
TableForm[t]
(* Clark Kimberling, May 01 2016 *)
Table[Reverse[Fibonacci[Range[n]]],{n,15}]//Flatten (* Harvey P. Dale, Jan 28 2019 *)

Formula

In every column, (1, 1, 2, 3, 5, ...); the nonzero Fibonacci numbers, A000045.

a(n,k) = A000045(n-k+1). - R. J. Mathar, Jun 23 2006

a(n) = A000045(m), where m = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012

Let P denote Pascal's triangle. Then P*A104762*P^(-1) = A121461. - Peter Bala, Apr 11 2013

a(n,k) = |round((r^n)*(s^k)/sqrt(5))|, where r = golden ratio = (1 + sqrt(5))/2, s = (1 - sqrt(5))/2, 1 <= k <= n-1, n >= 2. - Clark Kimberling, May 01 2016

G.f. of triangle: G(x,y) = x*y/((1-x-x^2)*(1-x*y)). - Robert Israel, May 01 2016

Extensions

Edited by N. J. A. Sloane at the suggestion of Philippe Deléham, Jun 11 2007

More terms from Philippe Deléham, Apr 21 2009