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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. 15
 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. 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 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. 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 Matthew Blair, Rigoberto Flórez, Antara Mukherjee, Matrices in the Hosoya triangle, arXiv:1808.05278 [math.CO], 2018. 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 R. Florez, R. A. Higuita, L Junes, GCD Property of the Generalized Star of David in the Generalized Hosoya Triangle, J. Int. Seq. 17 (2014) # 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, 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. 48 (2010), no. 2, 168-174. H. Hosoya, Fibonacci Triangle, The Fibonacci Quarterly, 14;2, 1976, 173-178. S. Klavzar, I. Peterin, Edge-counting vectors, Fibonacci cubes and Fibonacci triangle, 2005 preprint of Publ. Math. Debrecen 71/3-4 (2007), 267-278. 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 Initial rows of the triangle:   1   1 1   2 1 2   3 2 2 3   5 3 4 3 5   ... 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, ...   ... 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 CROSSREFS Cf. A000045, A003991, A001629, A098356, A283845. Sequence in context: A087061 A082860 A283845 * A174961 A104889 A290979 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 October 17 01:03 EDT 2019. Contains 328103 sequences. (Running on oeis4.)