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A058071 A Fibonacci triangle: triangle T(n,k) in which n-th row consists of the numbers F(k)F(n+2-k), where F() are the Fibonacci numbers, for n >= 0, 0<=k<=n+1. 13
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 (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.

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 F(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

REFERENCES

B. 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.

S. Klavzar, I. Peterin Edge-counting vectors, Fibonacci cubes and Fibonacci triangle, Publ. Math. Debrecen 71/3-4 (2007), 267-278.

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

C. Cobeli and A. Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98. - From N. J. A. Sloane, Feb 16 2013

Martin Griffiths, Digit Proportions in Zeckendorf Representations, Fibonacci Quart. 48 (2010), no. 2, 168-174.

H. Hosoya, Fibonacci Triangle, The Fibonacci Quarterly, 14;2, 1976, 173-178.

T. V. Trif, Solution to Problem 10706 proposed by J. G. Propp, Amer. Math. Monthly, 107 (Nov. 2000), p. 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

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

EXAMPLE

Rows 1,2,3,4,5:

1

1 1

2 1 2

3 2 2 3

5 3 4 3 5

MATHEMATICA

row[n_] := Table[Fibonacci[k]*Fibonacci[n-k+1], {k, 1, n}]; Table[row[n], {n, 1, 12}] // Flatten (* Jean-Fran├žois 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

CROSSREFS

Cf. A000045, A003991, A098356.

Sequence in context: A003984 A087061 A082860 * A174961 A104889 A117910

Adjacent sequences:  A058068 A058069 A058070 * A058072 A058073 A058074

KEYWORD

nonn,easy,tabl,nice

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

STATUS

approved

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Last modified December 3 12:46 EST 2016. Contains 278734 sequences.