OFFSET
0,4
COMMENTS
lim a(n)/a(n-1) = (1+sqrt(5))/2.
a(n-1) is the max coefficient in n-th Fibonacci polynomial (the polynomial F_0(x) is constant zero, and is not included in this sequence). - Vladimir Reshetnikov, Oct 09 2016
REFERENCES
Peter Boros (borospet(AT)freemail.hu): Lectures on Fibonacci's World at the SOTERIA Foundation, 1999.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..4793
Benjamin Aram Berendsohn, László Kozma, and Dániel Marx, Finding and counting permutations via CSPs, arXiv:1908.04673 [cs.DS], 2019.
Charles Bouillaguet, Boolean Polynomial Evaluation for the Masses, LIP6 Laboratory, Sorbonne Université (Paris, France) Cryptology ePrint Archive (2022) No. 1412.
S. M. Tanny and M. Zuker, On a unimodal sequence of binomial coefficients, Discrete Math. 9 (1974), 79-89.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
FORMULA
a(n) ~ 5^(1/4) * phi^(n+1) / sqrt(2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 09 2016
EXAMPLE
For n = 6, C(6,0) = 1, C(5,1) = 5, C(4,2) = 6, C(3,3) = 1. These binomial coefficients are the coefficients in the Fibonacci polynomial F_7(x) = x^6 + 5*x^4 + 6*x^2 + 1. The max coefficient is 6, so a(6) = 6.
MATHEMATICA
Table[Max[CoefficientList[Fibonacci[n + 1, x], x]], {n, 1, 30}] (* Vladimir Reshetnikov, Oct 07 2016 *)
PROG
(PARI) a(n)=my(k=(5*n-sqrtint(5*n^2+10*n+9)+6)\10); binomial(n-k, k) \\ Charles R Greathouse IV, Sep 22 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Miklos Kristof, Aug 22 2002
EXTENSIONS
a(0) = 1 prepended by Vladimir Reshetnikov, Oct 09 2016
STATUS
approved