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A058071
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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.
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12
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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; internal format)
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OFFSET
| 0,4
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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 (gerald.mcgarvey(AT)comcast.net), 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 (emanuele.munarini(AT)polimi.it), 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. [From Clark Kimberling, Jul 31 2011]
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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.
H. Hosoya, "Fibonacci Triangle", The Fibonacci Quarterly, 14;2, 1976, 173-178.
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.
T. V. Trif, Solution to Problem 10706 proposed by J. G. Propp, Amer. Math. Monthly, 107 (Nov. 2000), p. 866-867.
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LINKS
| Emanuele Munarini (emanuele.munarini(AT)polimi.it), Apr 01 2008, Table of n, a(n) for n = 0..860
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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 (emanuele.munarini(AT)polimi.it), Apr 01 2008
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EXAMPLE
| Rows 1,2,3,4,5:
1
1 1
2 1 2
3 2 2 3
5 3 4 3 5
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CROSSREFS
| Cf. A000045, A003991, A098356.
Sequence in context: A003984 A087061 A082860 * A174961 A104889 A117910
Adjacent sequences: A058068 A058069 A058070 * A058072 A058073 A058074
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KEYWORD
| nonn,easy,tabl,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 24 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Nov 27 2000
Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 15 2008 at the suggestion of R. J. Mathar
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