A175800 - OEIS

%I #15 May 12 2019 02:22:55

%S 0,0,0,0,0,0,1,1,1,1,1,2,0,0,2,0,1,1,1,1,0,2,2,0,0,1,1,1,1,1,2,0,0,0,

%T 0,1,1,1,1,2,4,2,0,0,1,1,3,1,1,2,2,0,0,2,1,1,3,1,1,4,2,2,2,1,1,3,1,1,

%U 0,2,0,2,2,3,1,1,1,1,2,2,0,2,2,1,1,1,1,2,2,4,0,0,1,1,1

%N Number of real zeros of the polynomial whose coefficients are the decimal digits of Fibonacci(n).

%C a(n) is the number of real zeros of the polynomial Sum_{k=0..p} d(k)*x^k

%C   where d(k) are the decimal digits of Fibonacci(n) = Sum_{i>=0} 10^i*d(i).

%e a(41) = 4 because Fibonacci(41) = 165580141 and the polynomial 1 + 4*x + x^2 + 8*x^4 + 5*x^5 + 5*x^6 + 6*x^7 + x^8 has 4 real roots, x0 = -5.160582776..., x2 = -1.173079878..., x3 = -0.7235395314..., and x4 = -0.2802116772...

%p A175800 := proc(n)

%p         d := convert(combinat[fibonacci](n),base,10) ;

%p         P := add( op(i,d)*x^(i-1),i=1..nops(d)) ;

%p         [fsolve(P,x,real)] ;

%p         nops(%) ;

%p end proc:

%p seq(A175800(n),n=1..45) ; # _R. J. Mathar_, Dec 06 2010

%Y Cf. A000045, A173667.

%K nonn,base

%O 1,12

%A _Michel Lagneau_, Dec 04 2010