|
%I #17 Jan 18 2022 04:08:51
%S 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,
%T 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,
%U 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
%N Decimal expansion of the constant term of the reduction of e^x by x^2->x+1.
%C 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
%C ...
%C c(0) + c(1)*x + c(2)*x^2 + ... ,
%C ...
%C 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):
%C ...
%C 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
%C ...
%C R(f(x)) = c(0) + c(1)*x + c(2)*(x+1) + c(3)*(2x+1) + c(4)(3x+2) + ..., so that
%C ...
%C R(f(x)) = sum{c(n)*(F(n)*x+F(n-1)}: n>=0}, where F=A000045 (Fibonacci sequence), so that
%C ...
%C 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.
%C 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.
%C e^x......... A193010 + x*A098689
%C e^(-x)...... A193026 + x*A099935
%C e^(2x)...... A193027 + x*A193028
%C e^(x/2)..... A193029 + x*A193030
%C sin x....... A193011 + x*A193012
%C cos x....... A193013 + x*A193014
%C sinh x...... A193015 + x*A193016
%C cosh x...... A193017 + x*A193025
%C 2^x......... A193031 + x*A193032
%C 2^(-x)...... A193009 + x*A193035
%C 3^x......... A193083 + x*A193084
%C t^x......... A193075 + x*A193076, t=(1+sqrt(5))/2
%C t^(-x)...... A193077 + x*A193078, t=(1+sqrt(5))/2
%C sinh(2x).... A193079 + x*A193080
%C cosh(2x).... A193081 + x*A193082
%C (e^x)cos x.. A193083 + x*A193084
%C (e^x)sin x.. A193085 + x*A193086
%C (cos x)^2... A193087 + x*A193088
%C (sin x)^2... A193089 + x*A193088
%F From _Amiram Eldar_, Jan 18 2022: (Start)
%F Equals 1 + Sum_{k>=1} Fibonacci(k-1)/k!.
%F Equals (sqrt(5)-1) * (2*sqrt(5)*exp(sqrt(5)) + 3*sqrt(5) + 5) / (20 * exp((sqrt(5)-1)/2)). (End)
%e 1.783922996312878767846236916090170972510...
%t f[x_] := Exp[x]; r[n_] := Fibonacci[n];
%t c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t u0 = N[Sum[c[n]*r[n - 1], {n, 0, 200}], 100]
%t RealDigits[u0, 10]
%Y Cf. A000045, A192232.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Jul 14 2011
|
