|
%I M2988 N1208 #139 Jun 25 2023 08:10:57
%S 1,3,15,60,260,1092,4641,19635,83215,352440,1493064,6324552,26791505,
%T 113490195,480752895,2036500788,8626757644,36543528780,154800876945,
%U 655747029795,2777789007071,11766903040368,49845401197200,211148507782800,894439432403425
%N Fibonomial coefficients: a(n) = F(n+1)*F(n+2)*F(n+3)/2, where F() = Fibonacci numbers A000045.
%C In a triangle having sides of F(n+1), 2*F(n+2) and F(n+3), the product of the area and circumradius will be a(n). For example: a triangle having sides of 5, 16 and 13 will have an area of 4*sqrt(51), a circumradius of 65*sqrt(51)/51, and the product is 4*65 = 260. - _Gary Detlefs_, Dec 14 2010
%C Explanation of this comment: if a triangle with sides (a, b, c) has a circumradius R and an area A, then A*R = abc/4; here, with a = F(n+1), b=2*F(n+2) and c=F(n+3), this gives a(n)= A*R. - _Bernard Schott_, Jan 26 2023
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001655/b001655.txt">Table of n, a(n) for n = 0..200</a>
%H A. Brousseau, <a href="http://www.fq.math.ca/Scanned/6-1/brousseau3.pdf">A sequence of power formulas</a>, Fib. Quart., 6 (1968), 81-83.
%H A. Brousseau, <a href="http://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972, p. 74.
%H Kenneth Edwards and Michael A. Allen, <a href="https://www.fq.math.ca/Papers1/58-5/edwards.pdf">A new combinatorial interpretation of the Fibonacci numbers cubed</a>, Fib. Q. 58:5 (2020) 128-134.
%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H David Treeby, <a href="https://www.fq.math.ca/Papers1/54-4/Treeby09132016.pdf">Further Physical Derivations of Fibonacci Summations</a>, Fibonacci Quart. 54 (2016), no. 4, 327-334.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-3,-1).
%F G.f.: 1/(1-3*x-6*x^2+3*x^3+x^4) = 1/((1+x-x^2)*(1-4*x-x^2)) (see Comments to A055870).
%F a(n) = A010048(n+3, 3) = fibonomial(n+3, 3).
%F a(n) = (1/2) * A065563(n).
%F a(n) = 4*a(n-1) + a(n-2) + ((-1)^n)*F(n+1), n >= 2; a(0)=1, a(1)=3.
%F a(n) = (F(n+3)^3 - F(n+2)^3 - F(n+1)^3)/6. - _Gary Detlefs_, Dec 24 2010
%F a(n-1) = Sum_{k=0..n} F(k+1)*F(k)^2, n >= 1. - _Wolfdieter Lang_, Aug 01 2012
%F From _Wolfdieter Lang_, Aug 09 2012: (Start)
%F a(n-1)*(-1)^n = Sum_{k=0..n} (-1)^k*F(k+1)^2*F(k), n >= 1. See the link under A215037, eq. (25).
%F a(n) = (F(3*(n+2)) + 2*(-1)^n*F(n+2))/10, n >= 0. See the same link, eq. (32). (End)
%F a(n) = -a(-4-n)*(-1)^n for all n in Z. - _Michael Somos_, Sep 19 2014
%F 0 = a(n)*(-a(n+1) - a(n+2)) + a(n+1)*(-3*a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Sep 19 2014
%F O.g.f.: exp( Sum_{n >= 1} L(n)*L(2*n)*x^n/n ), where L(n) = A000032(n) is a Lucas number. Cf. A114525, A256178. - _Peter Bala_, Mar 18 2015
%F Sum_{n>=0} (-1)^n/a(n) = 2 * A079586 - 6. - _Amiram Eldar_, Oct 04 2020
%F The formula by Gary Detlefs above is valid for all sequences of the Fibonacci type f(n) = f(n-1) + f(n-2): 3*f(n+2)*f(n+1)*f(n) = f(n+2)^3 - f(n+1)^3 - f(n)^3. - _Klaus Purath_, Mar 25 2021
%F a(n) = sqrt(Sum_{j=1..n+1} F(j)^3*F(j+1)^3). See Treeby link. - _Michel Marcus_, Apr 10 2022
%F a(n) = Sum_{k=1..n+1} A000129(k)*A056570(n+2-k). - _Michael A. Allen_, Jan 25 2023
%e G.f. = 1 + 3*x + 15*x^2 + 60*x^3 + 260*x^4 + 1092*x^5 + 4641*x^6 + ...
%p A001655:=1/(z**2-z-1)/(z**2+4*z-1); # _Simon Plouffe_ in his 1992 dissertation.
%t Table[(Fibonacci[n+3]*Fibonacci[n+2]*Fibonacci[n+1])/2, {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 23 2009 *)
%t LinearRecurrence[{3, 6, -3, -1}, {1, 3, 15, 60}, 25] (* _Jean-François Alcover_, Sep 23 2017 *)
%o (PARI) b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j)); vector(20, n, b(n-1, 3)) \\ _Joerg Arndt_, May 08 2016
%o (Magma) [Fibonacci(n+3)*Fibonacci(n+2)*Fibonacci(n+1)/2: n in [0..30]]; // _Vincenzo Librandi_, May 09 2016
%Y Cf. A066258 (first differences), A215037 (partial sums), A363753 (alternating sums).
%Y Cf. A000045, A055870, A065563, A079586, A114525, A256178.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
|
