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A175830 Number of real-valued zeros of the polynomial whose coefficients are the leading n+1 decimal digits of Euler's constant, A011543(n). 0
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, 2, 1, 2, 1, 2, 1, 2, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..42.

EXAMPLE

a(0) = 0 because 2 => P(0,x)=2 is a constant and has no real root.

a(2) = 2  because 271 => P(2,x) = 1+7x + 2x^2 has 2 real roots.

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.

MAPLE

with(numtheory):Digits:=50: T:=array(1..45):for zz from 0 to 43 do:n:=floor(exp(1)*10^zz):

  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,

  10):v:=iquo(q, 10):n0:=v :u: T[m]:=u:od: x:=fsolve(T[1]+ T[2]*z + T[3]*z^2+

  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+

  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: ~

CROSSREFS

Cf. A173667, A175799, A001113, A011543.

Sequence in context: A161060 A161264 A161289 * A161316 A161250 A161035

Adjacent sequences:  A175827 A175828 A175829 * A175831 A175832 A175833

KEYWORD

nonn,base,less

AUTHOR

Michel Lagneau, Dec 05 2010

STATUS

approved

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Last modified October 20 13:38 EDT 2021. Contains 348108 sequences. (Running on oeis4.)