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A193010
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Decimal expansion of the constant term of the reduction of e^x by x^2->x+1.
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36
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1, 7, 8, 3, 9, 2, 2, 9, 9, 6, 3, 1, 2, 8, 7, 8, 7, 6, 7, 8, 4, 6, 2, 3, 6, 9, 1, 6, 0, 9, 0, 1, 7, 0, 9, 7, 2, 5, 1, 0, 2, 9, 8, 6, 0, 6, 3, 3, 8, 4, 1, 2, 1, 7, 8, 7, 0, 7, 0, 0, 0, 7, 3, 6, 6, 8, 9, 5, 2, 5, 9, 7, 4, 0, 0, 2, 0, 3, 0, 2, 5, 3, 5, 4, 8, 2, 6, 1, 5, 6, 5, 0, 5, 6, 7, 1, 9, 4, 5, 2
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OFFSET
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1,2
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COMMENTS
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Suppose that q and s are polynomials and degree(q)>degree(s). The reduction of a polynomial p by q->s is introduced at A192232. If p is replaced by a function f having power series
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c(0) + c(1)*x + c(2)*x^2 + ... ,
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then the reduction, R(f), of f by q->s is here introduced as the limit, if it exists, of the reduction of p(n,x) by q->s, where p(n,x) is the n-th partial sum of f(x):
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R(f(x)) = c(0)*R(1) + c(1)*R(x) + c(2)*R(x^2) + ... If q(x)=x^2 and s(x)=x+1, then
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R(f(x)) = c(0) + c(1)*x + c(2)*(x+1) + c(3)*(2x+1) + c(4)(3x+2) + ..., so that
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R(f(x)) = sum{c(n)*(F(n)*x+F(n-1)}: n>=0}, where F=A000045 (Fibonacci sequence), so that
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R(f(x)) = u0 + x*u1 where u0=sum{c(n)F(n-1): n>=0}, u1=sum(c(n)F(n): n>=0); the numbers u0 and u1 are given by A193010 and A098689.
Following is a list of reductions by x^2->x+1 of selected functions. Each sequence A-number refers to the constant represented by the sequence. Adjustments for offsets are needed in some cases.
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LINKS
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FORMULA
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Equals 1 + Sum_{k>=1} Fibonacci(k-1)/k!.
Equals (sqrt(5)-1) * (2*sqrt(5)*exp(sqrt(5)) + 3*sqrt(5) + 5) / (20 * exp((sqrt(5)-1)/2)). (End)
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EXAMPLE
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1.783922996312878767846236916090170972510...
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MATHEMATICA
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f[x_] := Exp[x]; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u0 = N[Sum[c[n]*r[n - 1], {n, 0, 200}], 100]
RealDigits[u0, 10]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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