A073028 - OEIS

%I #38 Oct 27 2022 09:34:58

%S 1,1,1,2,3,4,6,10,15,21,35,56,84,126,210,330,495,792,1287,2002,3003,

%T 5005,8008,12376,19448,31824,50388,77520,125970,203490,319770,497420,

%U 817190,1307504,2042975,3268760,5311735,8436285,13123110,21474180,34597290

%N a(n) = max{ C(n,0), C(n-1,1), C(n-2,2), ..., C(n-n,n) }.

%C lim a(n)/a(n-1) = (1+sqrt(5))/2.

%C 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

%D Peter Boros (borospet(AT)freemail.hu): Lectures on Fibonacci's World at the SOTERIA Foundation, 1999.

%H Charles R Greathouse IV, <a href="/A073028/b073028.txt">Table of n, a(n) for n = 0..4793</a>

%H Benjamin Aram Berendsohn, László Kozma, and Dániel Marx, <a href="https://arxiv.org/abs/1908.04673">Finding and counting permutations via CSPs</a>, arXiv:1908.04673 [cs.DS], 2019.

%H Charles Bouillaguet, <a href="https://eprint.iacr.org/2022/1412.pdf">Boolean Polynomial Evaluation for the Masses</a>, LIP6 Laboratory, Sorbonne Université (Paris, France) Cryptology ePrint Archive (2022) No. 1412.

%H S. M. Tanny and M. Zuker, <a href="http://dx.doi.org/10.1016/0012-365X(74)90073-9">On a unimodal sequence of binomial coefficients</a>, Discrete Math. 9 (1974), 79-89.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>.

%F a(n) = binomial(n-A060065(n), A060065(n)). - _Vladeta Jovovic_, Jun 16 2004

%F 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

%e 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.

%t Table[Max[CoefficientList[Fibonacci[n + 1, x], x]], {n, 1, 30}] (* _Vladimir Reshetnikov_, Oct 07 2016 *)

%o (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

%Y Cf. A060065, A277282, A168561.

%K easy,nonn

%O 0,4

%A _Miklos Kristof_, Aug 22 2002

%E a(0) = 1 prepended by _Vladimir Reshetnikov_, Oct 09 2016