|
| |
|
|
A058071
|
|
A Fibonacci triangle: triangle T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1), for n >= 0, 0 <= k <= n.
|
|
17
|
|
|
|
1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 4, 3, 5, 8, 5, 6, 6, 5, 8, 13, 8, 10, 9, 10, 8, 13, 21, 13, 16, 15, 15, 16, 13, 21, 34, 21, 26, 24, 25, 24, 26, 21, 34, 55, 34, 42, 39, 40, 40, 39, 42, 34, 55, 89, 55, 68, 63, 65, 64, 65, 63, 68, 55, 89, 144, 89, 110, 102, 105, 104, 104, 105, 102, 110, 89, 144
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Or, multiplication table of the positive Fibonacci numbers read by antidiagonals.
Or, triangle of products of nonzero Fibonacci numbers.
Or, a two-dimensional square Fibonacci array read by antidiagonals, with offset 1: T(1,1) = T(1,2) = T(2,1) = T(2,2) = 1; thereafter T(m,n) = max {T(m,n-2) + T(m,n-1), T(m-2,n) + T(m-1,n), T(m-2,n-2) + T(m-1,n-1)}. If "max" is changed to "min" we get A283845. - N. J. A. Sloane, Mar 31 2017
Row sums are A001629 (Fibonacci numbers convolved with themselves.). The main diagonal and first subdiagonal are Fibonacci numbers, for other entries T(n,k) = T(n-1,k) + T(n-2,k). The central numbers form A006498. - Gerald McGarvey, Jun 02 2005
Alternating row sums = (1, 0, 3, 0, 8, ...), given by Fibonacci(2n) if n even, else zero.
Row n = edge-counting vector for the Fibonacci cube F(n+1) embedded in the natural way in the hypercube Q(n+1). - Emanuele Munarini, Apr 01 2008
T(n,k) = number of appearances of a(k) in p(n) in the n-th convergent p(n)/q(n) of the formal infinite continued fraction [a(0), a(1), ...]; e.g., p(3) = a(0)*a(1)*a(2)*a(3) + a(0)*a(1) + a(0)*a(3) + a(2)*a(3) + 1. Also, T(n,k) = number of appearances of a(k+1) in q(n+1); e.g., q(3) = a(1)*a(2)*a(3) + a(1) + a(3). - Clark Kimberling, Dec 21 2015
Each row is a palindrome, and the central term of row 2n is the square of the F(n+1), where F = A000045 (Fibonacci numbers). - Clark Kimberling, Dec 21 2015
Also called Hosoya's triangle, after the Japanese chemist Haruo Hosoya (b. 1936). - Amiram Eldar, Jun 10 2021
|
|
|
REFERENCES
|
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 27.
Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Chap. 15, Hosoya's Triangle, Wiley, New York, 2001.
|
|
|
LINKS
|
Haruo Hosoya, Fibonacci Triangle, The Fibonacci Quarterly, Vol. 14, No. 2 (1976), pp. 173-178.
|
|
|
FORMULA
|
Row n: F(1)*F(n), F(2)*F(n-1), ..., F(n)*F(1).
G.f.: T(x,y) = 1/((1-x-x^2)(1-xy-x^2y^2)). Recurrence: T(n+4,k+2) = T(n+3,k+2) + T(n+3,k+1) + T(n+2,k+2) - T(n+2,k+1) + T(n+2,k) - T(n+1,k+1) - T(n+1,k) - T(n,k). - Emanuele Munarini, Apr 01 2008
Column k is the (generalized) Fibonacci sequence having first two terms F(k+1), F(k+1). - Clark Kimberling, Dec 21 2015
T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1).
Sum_{k=0..n} T(n, k) = A001629(n+2).
Sum_{k=0..floor(n/2)} T(n, k) = A024458(n+1).
Sum_{k=1..n-1} T(n, k) = A004798(n-1), n >= 2.
Sum_{k=0..floor(n/2)} T(n-k, k) = A250111(n+2).
T(n, n-k) = T(n, k). (End)
|
|
|
EXAMPLE
|
Triangle begins as:
1;
1, 1;
2, 1, 2;
3, 2, 2, 3;
5, 3, 4, 3, 5;
8, 5, 6, 6, 5, 8;
13, 8, 10, 9, 10, 8, 13;
21, 13, 16, 15, 15, 16, 13, 21;
34, 21, 26, 24, 25, 24, 26, 21, 34;
...
As a square array:
1, 1, 2, 3, 5, 8, 13, 21, ...
1, 1, 2, 3, 5, 8, 13, 21, ...
2, 2, 4, 6, 10, 16, 26, ...
3, 3, 6, 9, 15, 24, ...
5, 5, 10, 15, 25, ...
8, 8, 16, 24, ...
13, 13, 26, ...
21, 21, ...
|
|
|
MATHEMATICA
|
row[n_] := Table[Fibonacci[k]*Fibonacci[n-k+1], {k, 1, n}]; Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)
|
|
|
PROG
|
(Haskell)
a058071 n k = a058071_tabl !! n !! k
a058071_row n = a058071_tabl !! n
a058071_tabl = map (\fs -> zipWith (*) fs $ reverse fs) a104763_tabl
(Magma) [Fibonacci(k+1)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2022
(SageMath) flatten([[fibonacci(k+1)*fibonacci(n-k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 06 2022
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
