A175800 Number of real zeros of the polynomial whose coefficients are the decimal digits of Fibonacci(n).

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, 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, 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

Offset: 1

Author

Michel Lagneau, Dec 04 2010

Keywords

• nonn
• base

Comments

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

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

Examples

			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...
		

Crossrefs

Cf. A000045, A173667.

Programs

Maple

A175800 := proc(n)
        d := convert(combinat[fibonacci](n),base,10) ;
        P := add( op(i,d)*x^(i-1),i=1..nops(d)) ;
        [fsolve(P,x,real)] ;
        nops(%) ;
end proc:
seq(A175800(n),n=1..45) ; # R. J. Mathar, Dec 06 2010