A193009 -id:A193009 - cp's OEIS frontend

A193010 Decimal expansion of the constant term of the reduction of e^x by x^2->x+1.

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

Offset: 1

Clark Kimberling, Jul 14 2011

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(0) + c(1)*x + c(2)*x^2 + ... ,
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):
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
R(f(x)) = c(0) + c(1)*x + c(2)*(x+1) + c(3)*(2x+1) + c(4)(3x+2) + ..., so that
R(f(x)) = Sum_{n>=0} c(n)*(F(n)*x+F(n-1)), where F=A000045 (Fibonacci sequence), so that
R(f(x)) = u0 + x*u1 where u0 = Sum_{n>=0} c(n)*F(n-1), u1 = Sum_{n>=0} c(n)*F(n); 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.
e^x......... A193010 + x*A098689
e^(-x)...... A193026 + x*A099935
e^(2x)...... A193027 + x*A193028
e^(x/2)..... A193029 + x*A193030
sin x....... A193011 + x*A193012
cos x....... A193013 + x*A193014
sinh x...... A193015 + x*A193016
cosh x...... A193017 + x*A193025
2^x......... A193031 + x*A193032
2^(-x)...... A193009 + x*A193035
3^x......... A193083 + x*A193084
t^x......... A193075 + x*A193076, t=(1+sqrt(5))/2
t^(-x)...... A193077 + x*A193078, t=(1+sqrt(5))/2
sinh(2x).... A193079 + x*A193080
cosh(2x).... A193081 + x*A193082
(e^x)cos x.. A193083 + x*A193084
(e^x)sin x.. A193085 + x*A193086
(cos x)^2... A193087 + x*A193088
(sin x)^2... A193089 + x*A193088

			1.783922996312878767846236916090170972510...

Cf. A000045, A192232.

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]

From Amiram Eldar, Jan 18 2022: (Start)
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)

A193008 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 2, 10, 31, 78, 170, 339, 636, 1144, 1997, 3412, 5740, 9549, 15758, 25854, 42243, 68818, 111878, 181615, 294520, 477276, 773057, 1251720, 2026296, 3279673, 5307770, 8589394, 13899271, 22490934, 36392642, 58886187, 95281620, 154170784

Offset: 0

Clark Kimberling, Jul 14 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+1+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A192232, A192744, A192951, A193009.

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^3 + 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A193008 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193009 *)

a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).

G.f.: (7*x^2-2*x+1)/((x-1)^3*(x^2+x-1)). [Colin Barker, Nov 12 2012]

A193035 Decimal expansion of the coefficient of x in the reduction of 2^(-x) by x^2->x+1.

Original entry on oeis.org

5, 4, 0, 6, 8, 2, 6, 4, 1, 9, 5, 8, 4, 8, 0, 3, 8, 3, 7, 7, 7, 4, 1, 0, 5, 5, 2, 7, 2, 4, 2, 2, 1, 3, 0, 1, 2, 4, 8, 5, 3, 2, 6, 9, 1, 1, 1, 1, 6, 8, 3, 2, 4, 5, 8, 9, 2, 4, 2, 2, 0, 4, 6, 0, 0, 1, 1, 2, 4, 2, 6, 6, 3, 6, 2, 3, 0, 3, 2, 9, 8, 4, 8, 6, 1, 1, 9, 1, 3, 0, 5, 0, 8, 7, 2, 7, 3, 3, 7, 2, 6, 3

Offset: 0

Clark Kimberling, Jul 14 2011

Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.

			-0.540682641958480383777410552724221301248532691111...

Cf. A000045, A001622, A193010, A192232, A193009.

f[x_] := 2^(-x); r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]

From Amiram Eldar, Jan 19 2022: (Start)
Equals Sum_{k>=0} (-log(2))^k*Fibonacci(k)/k!.
Equals -(2^sqrt(5) - 1)/(sqrt(5)*2^phi), where phi is the golden ratio (A001622). (End)

a(99) corrected by Georg Fischer, Aug 04 2024

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A193010 Decimal expansion of the constant term of the reduction of e^x by x^2->x+1.

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A193008 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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A193035 Decimal expansion of the coefficient of x in the reduction of 2^(-x) by x^2->x+1.

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