A058071 A Fibonacci triangle: triangle T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1), for n >= 0, 0 <= k <= n.

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

Offset: 0

Author

N. J. A. Sloane, Nov 24 2000

Keywords

• nonn
• easy
• tabl
• nice

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

Examples

			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, ...
		

References

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
• 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.
• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 27.
• 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.

Crossrefs

Cf. A000045, A003991, A001629, A098356, A283845.

Cf. A001654, A004798, A007598, A024458, A250111

Programs

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

Magma

[Fibonacci(k+1)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2022

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 *)

PARI

T(n,k)=fibonacci(k)*fibonacci(n+2-k) \\ Charles R Greathouse IV, Feb 07 2017

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

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

T(n,k) = A104763(n+1,k+1) * A104763(n+1,n+1-k). - Reinhard Zumkeller, Aug 15 2013

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)

Extensions

More terms from James 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