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
The augmentation of A058071 is the triangle A193595. To fit the definition of augmented triangle at A103091, it is helpful to represent A058071 using p(n,k)=F(k+1)*F(n+1-k) for 0<=k<=n. - Clark Kimberling, Jul 31 2011
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
Emanuele Munarini and Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Arthur T. Benjamin and Daniela Elizondo, Counting on Hosoya's Triangle, Fibonacci Quart. 60 (2022), no. 5, 47-55.
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Matrices in the Hosoya triangle, arXiv:1808.05278 [math.CO], 2018.
Hsin-Yun Ching, Rigoberto Flórez, and Antara Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56 (104), No. 1 (2013), pp. 73-98. - From N. J. A. Sloane, Feb 16 2013
Rigoberto Flórez, Robinson A. Higuita and Leandro Junes, GCD Property of the Generalized Star of David in the Generalized Hosoya Triangle, J. Int. Seq., Vol. 17 (2014), Article 14.3.6.
Rigoberto Florez, Robinson A. Higuita, Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, arXiv:1706.04247 [math.CO], 2017.
Rigoberto Flórez, Robinson A. Higuita, and Antara Mukherjee, The Geometry of some Fibonacci Identities in the Hosoya Triangle, arXiv:1804.02481 [math.NT], 2018.
Martin Griffiths, Digit Proportions in Zeckendorf Representations, Fibonacci Quart., Vol. 48, No. 2 (2010), pp. 168-174.
Haruo Hosoya, Fibonacci Triangle, The Fibonacci Quarterly, Vol. 14, No. 2 (1976), pp. 173-178.
Sandi Klavžar and Iztok Peterin, Edge-counting vectors, Fibonacci cubes and Fibonacci triangle, 2005 preprint of Publ. Math. Debrecen, Vol. 71, No. 3-4 (2007), pp. 267-278.
Tiberiu V. Trif, Solution to Problem 10706 proposed by J. G. Propp, Amer. Math. Monthly, Vol. 107, No. 9 (Nov. 2000), pp. 866-867.
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
From G. C. Greubel, Apr 06 2022: (Start)
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, 0) = A000045(n+1).
T(2*n, n) = A007598(n+1).
T(2*n+1, n) = A001654(n+1).
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
-- Reinhard Zumkeller, Aug 15 2013
(PARI) T(n, k)=fibonacci(k)*fibonacci(n+2-k) \\ Charles R Greathouse IV, Feb 07 2017
(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
N. J. A. Sloane, Nov 24 2000
EXTENSIONS
More terms from James A. Sellers, Nov 27 2000
Edited by N. J. A. Sloane, Sep 15 2008 at the suggestion of R. J. Mathar
Name edited by G. C. Greubel, Apr 06 2022
STATUS
approved