A099935 -id:A099935 - 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)

A098689 Decimal expansion of Sum_{n>=0} Fibonacci(n)/n!.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Oct 27 2004

			2.01432273345831573658134625546975913565911146958112...

Cf. A000045, A001113, A099935.

with(combinat): evalf(sum(fibonacci(n)/n!, n=0..infinity), 120);

RealDigits[ Sum[ Fibonacci[n]/n!, {n, 0, 80}], 10, 105][[1]] (* Robert G. Wilson v, Nov 02 2004 *)
(E^GoldenRatio - E^(1-GoldenRatio))/(2*GoldenRatio-1) // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 13 2013 *)

sumpos(n=0, fibonacci(n)/n!) \\ Michel Marcus, Feb 07 2022

Equals (2/sqrt(5))*exp(1/2)*sinh(sqrt(5)/2). - Vladeta Jovovic, Oct 30 2004

Equals e * A099935. - Amiram Eldar, Feb 07 2022

More terms from Robert G. Wilson v, Nov 02 2004

A193026 Decimal expansion of the constant term of the reduction of e^(-x) by x^2->x+1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2011

Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010. The coefficient of x in this reduction is the constant at A099935.

			1.39729651650004415809334932390899486052640...

Cf. A000045, A193010, A192232, A099935.

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, 100}], 100]
RealDigits[u0, 10]

From Amiram Eldar, Jan 18 2022: (Start)
Equals 1 + Sum_{k>=1} (-1)^k*Fibonacci(k-1)/k!.
Equals exp(-1/2)*(1 + sqrt(5)/5 + 2/(exp(sqrt(5))-1))*sinh(sqrt(5)/2). (End)

A328495 Decimal expansion of Sum_{k>=0} (-1)^k*L(k)/k!, where L(k) is the k-th Lucas number (A000032).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 22 2019

			2.053565111476510960344914661146965309320258644945918...

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd edition, Wiley, 2017, chapter 13.8, pp. 248-250.

Cf. A000032, A001113, A001622, A099935, A139341, A139342, A328344, A249455.

Digits := 100: 2*exp(-1/2)*cosh(sqrt(5)/2)*10^86:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 22 2019

RealDigits[Exp[-GoldenRatio] + Exp[GoldenRatio - 1], 10, 100][[1]]

Equals exp(-phi) + exp(phi-1), where phi is the golden ratio (A001622).
Equals 2*exp(-1/2)*cosh(sqrt(5)/2) = A249455*cosh(phi - 1/2). - Peter Luschny, Oct 22 2019
Equals A328344 / e. - Amiram Eldar, Feb 06 2022

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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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A098689 Decimal expansion of Sum_{n>=0} Fibonacci(n)/n!.

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A193026 Decimal expansion of the constant term of the reduction of e^(-x) by x^2->x+1.

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A328495 Decimal expansion of Sum_{k>=0} (-1)^k*L(k)/k!, where L(k) is the k-th Lucas number (A000032).

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