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%I #9 Mar 08 2015 18:14:09
%S 0,1,2,1,2,1,2,1,2,1,2,1,2,3,2,1,2,1,2,1,2,3,2,1,2,1,2,3,2,3,2,1,2,3,
%T 2,1,2,1,2,1,2,1,2
%N Number of real-valued zeros of the polynomial whose coefficients are the leading n+1 decimal digits of Euler's constant, A011543(n).
%C a(n) is the number of real-valued zeros of the polynomial P(n,x) = sum_{k=0..n} e(k) x^k where e = 2.7182818... and A011543(n) = sum_{k>=0} e(k)*10^k.
%e a(0) = 0 because 2 => P(0,x)=2 is a constant and has no real root.
%e a(2) = 2 because 271 => P(2,x) = 1+7x + 2x^2 has 2 real roots.
%e a(13) = 3 because 27182818284590 => P(13,x) = 9x +5x^2 +4x^3 +8x^4 +2x^5 +8x^6 +x^7 +8x^8 +2x^9 +8x^10 +x^11 +7x^12 +2x^13 has 3 real roots, -3.664218401…, -0.7829315178… and 0.
%p with(numtheory):Digits:=50: T:=array(1..45):for zz from 0 to 43 do:n:=floor(exp(1)*10^zz):
%p for i from 1 to 43 do: T[i]:=0:od: l:=length(n) : n0:=n:for m from 1 to l do:q:=n0:u:=irem(q,
%p 10):v:=iquo(q, 10):n0:=v :u: T[m]:=u:od: x:=fsolve(T[1]+ T[2]*z + T[3]*z^2+
%p T[4]*z^3+ T[5]*z^4 + T[6]*z^5 + T[7]*z^6 + T[8]*z^7 + T[9]*z^8 + T[10]*z^9+
%p T[11]*z^10+ T[12]*z^11 + T[13]*z^12 + T[14]*z^13 + T[15]*z^14+ T[16]*z^15+ T[17]*z^16 + T[18]*z^17 + T[19]*z^18 + T[20]*z^19 + T[21]*z^20 + T[22]*z^21+ T[23]*z^22+ T[24]*z^23 + T[25]*z^24 + T[26]*z^25+ T[27]*z^26+ T[28]*z^27+ T[29]*z^28 + T[30]*z^29 + T[31]*z^30+ T[32]*z^31 + T[33]*z^32 + T[34]*z^33+ T[35]*z^34+ T[36]*z^35 + T[37]*z^36 + T[38]*z^37+ T[39]*z^38 + T[40]*z^39+ T[41]*z^40+ T[42]*z^41 + T[43]*z^42, z, real):x1:=[x]: x2:=nops(x1): printf ( "%d %d %d\n",zz,n,x2):od: ~
%Y Cf. A173667, A175799, A001113, A011543.
%K nonn,base,less
%O 0,3
%A _Michel Lagneau_, Dec 05 2010
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