This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A058071 #82 Jul 02 2025 16:02:00
%S A058071 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,
%T A058071 15,15,16,13,21,34,21,26,24,25,24,26,21,34,55,34,42,39,40,40,39,42,34,
%U A058071 55,89,55,68,63,65,64,65,63,68,55,89,144,89,110,102,105,104,104,105,102,110,89,144
%N A058071 A Fibonacci triangle: triangle T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1), for n >= 0, 0 <= k <= n.
%C A058071 Or, multiplication table of the positive Fibonacci numbers read by antidiagonals.
%C A058071 Or, triangle of products of nonzero Fibonacci numbers.
%C A058071 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
%C A058071 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
%C A058071 Alternating row sums = (1, 0, 3, 0, 8, ...), given by Fibonacci(2n) if n even, else zero.
%C A058071 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
%C A058071 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
%C A058071 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
%C A058071 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
%C A058071 Also called Hosoya's triangle, after the Japanese chemist Haruo Hosoya (b. 1936). - _Amiram Eldar_, Jun 10 2021
%D A058071 Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%D A058071 Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Chap. 15, Hosoya's Triangle, Wiley, New York, 2001.
%H A058071 Emanuele Munarini and Reinhard Zumkeller, <a href="/A058071/b058071.txt">Rows n = 0..120 of table, flattened</a>
%H A058071 Arthur T. Benjamin and Daniela Elizondo, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/60-5/benjamin.pdf">Counting on Hosoya's Triangle</a>, Fibonacci Quart. 60 (2022), no. 5, 47-55.
%H A058071 Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, <a href="https://arxiv.org/abs/1808.05278">Matrices in the Hosoya triangle</a>, arXiv:1808.05278 [math.CO], 2018.
%H A058071 Boris A. Bondarenko, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a>, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 27.
%H A058071 Hsin-Yun Ching, Rigoberto Flórez, and Antara Mukherjee, <a href="https://arxiv.org/abs/2009.02770">Families of Integral Cographs within a Triangular Arrays</a>, arXiv:2009.02770 [math.CO], 2020.
%H A058071 Cristian Cobeli and Alexandru Zaharescu, <a href="http://rms.unibuc.ro/bulletin/pdf/56-1/PromenadePascalPart1.pdf">Promenade around Pascal Triangle-Number Motives</a>, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56 (104), No. 1 (2013), pp. 73-98. - From _N. J. A. Sloane_, Feb 16 2013
%H A058071 Rigoberto Flórez, Robinson A. Higuita and Leandro Junes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Florez/florez3.html">GCD Property of the Generalized Star of David in the Generalized Hosoya Triangle</a>, J. Int. Seq., Vol. 17 (2014), Article 14.3.6.
%H A058071 Rigoberto Florez, Robinson A. Higuita, Antara Mukherjee, <a href="https://arxiv.org/abs/1706.04247">Star of David and other patterns in the Hosoya-like polynomials triangles</a>, arXiv:1706.04247 [math.CO], 2017.
%H A058071 Rigoberto Flórez, Robinson A. Higuita, and Antara Mukherjee, <a href="https://arxiv.org/abs/1804.02481">The Geometry of some Fibonacci Identities in the Hosoya Triangle</a>, arXiv:1804.02481 [math.NT], 2018.
%H A058071 Martin Griffiths, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/48-2/Griffiths.pdf">Digit Proportions in Zeckendorf Representations</a>, Fibonacci Quart., Vol. 48, No. 2 (2010), pp. 168-174.
%H A058071 Haruo Hosoya, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-2/hosoya.pdf">Fibonacci Triangle</a>, The Fibonacci Quarterly, Vol. 14, No. 2 (1976), pp. 173-178.
%H A058071 Sandi Klavžar and Iztok Peterin, <a href="https://www.fmf.uni-lj.si/~klavzar/preprints/projections-final.pdf">Edge-counting vectors, Fibonacci cubes and Fibonacci triangle</a>, 2005 preprint of Publ. Math. Debrecen, Vol. 71, No. 3-4 (2007), pp. 267-278.
%H A058071 Tiberiu V. Trif, <a href="http://www.jstor.org/stable/2695753">Solution to Problem 10706 proposed by J. G. Propp</a>, Amer. Math. Monthly, Vol. 107, No. 9 (Nov. 2000), pp. 866-867.
%F A058071 Row n: F(1)*F(n), F(2)*F(n-1), ..., F(n)*F(1).
%F A058071 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
%F A058071 T(n,k) = A104763(n+1,k+1) * A104763(n+1,n+1-k). - _Reinhard Zumkeller_, Aug 15 2013
%F A058071 Column k is the (generalized) Fibonacci sequence having first two terms F(k+1), F(k+1). - _Clark Kimberling_, Dec 21 2015
%F A058071 From _G. C. Greubel_, Apr 06 2022: (Start)
%F A058071 T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1).
%F A058071 Sum_{k=0..n} T(n, k) = A001629(n+2).
%F A058071 Sum_{k=0..floor(n/2)} T(n, k) = A024458(n+1).
%F A058071 Sum_{k=1..n-1} T(n, k) = A004798(n-1), n >= 2.
%F A058071 Sum_{k=0..floor(n/2)} T(n-k, k) = A250111(n+2).
%F A058071 T(n, 0) = A000045(n+1).
%F A058071 T(2*n, n) = A007598(n+1).
%F A058071 T(2*n+1, n) = A001654(n+1).
%F A058071 T(n, n-k) = T(n, k). (End)
%e A058071 Triangle begins as:
%e A058071 1;
%e A058071 1, 1;
%e A058071 2, 1, 2;
%e A058071 3, 2, 2, 3;
%e A058071 5, 3, 4, 3, 5;
%e A058071 8, 5, 6, 6, 5, 8;
%e A058071 13, 8, 10, 9, 10, 8, 13;
%e A058071 21, 13, 16, 15, 15, 16, 13, 21;
%e A058071 34, 21, 26, 24, 25, 24, 26, 21, 34;
%e A058071 ...
%e A058071 As a square array:
%e A058071 1, 1, 2, 3, 5, 8, 13, 21, ...
%e A058071 1, 1, 2, 3, 5, 8, 13, 21, ...
%e A058071 2, 2, 4, 6, 10, 16, 26, ...
%e A058071 3, 3, 6, 9, 15, 24, ...
%e A058071 5, 5, 10, 15, 25, ...
%e A058071 8, 8, 16, 24, ...
%e A058071 13, 13, 26, ...
%e A058071 21, 21, ...
%t A058071 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 *)
%o A058071 (Haskell)
%o A058071 a058071 n k = a058071_tabl !! n !! k
%o A058071 a058071_row n = a058071_tabl !! n
%o A058071 a058071_tabl = map (\fs -> zipWith (*) fs $ reverse fs) a104763_tabl
%o A058071 -- _Reinhard Zumkeller_, Aug 15 2013
%o A058071 (PARI) T(n,k)=fibonacci(k)*fibonacci(n+2-k) \\ _Charles R Greathouse IV_, Feb 07 2017
%o A058071 (Magma) [Fibonacci(k+1)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 06 2022
%o A058071 (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
%Y A058071 Cf. A000045, A003991, A001629, A098356, A283845.
%Y A058071 Cf. A001654, A004798, A007598, A024458, A250111
%K A058071 nonn,easy,tabl,nice
%O A058071 0,4
%A A058071 _N. J. A. Sloane_, Nov 24 2000
%E A058071 More terms from _James Sellers_, Nov 27 2000
%E A058071 Edited by _N. J. A. Sloane_, Sep 15 2008 at the suggestion of _R. J. Mathar_
%E A058071 Name edited by _G. C. Greubel_, Apr 06 2022