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

1,4

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

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.

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

Example

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)

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 *)

Crossrefs

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

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

Sequence in context: A194543 A287920 A027293 * A152462 A180360 A318805

Adjacent sequences: A104759 A104760 A104761 * A104763 A104764 A104765

Keyword

nonn,easy,tabl

Author

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

Extensions

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

More terms from Philippe Deléham, Apr 21 2009

Status

approved