login
A104762
Triangle read by rows: row n contains first n nonzero Fibonacci numbers in decreasing order.
10
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
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
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