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A175834
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Number of real zeros of the polynomial whose coefficients are the decimal expansion of the golden ratio truncated to n places (A011551).
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1
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0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2
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OFFSET
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0,3
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COMMENTS
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a(n) = number of real zeros of the polynomial P(n,x) = sum_{k=0..n} p(k) x^k where p(k) are the digits of the decimal expansion of floor(GoldenRatio *10^n) and GoldenRatio = 1.6180339 ....
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LINKS
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EXAMPLE
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a(4) = 2 because 16180 => P(4,x) = 8x+x^2+6x^3+x^4 has 2 real roots :
x0= - 6.053134348… and x1 = 0.
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MAPLE
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with(numtheory):Digits:=50: T:=array(1..45):for zz from 0 to 43 do:n:=floor(((1+sqrt(5))/2)*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:
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CROSSREFS
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KEYWORD
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nonn,base,less
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AUTHOR
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STATUS
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approved
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