The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A193029 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A193029

Decimal expansion of the constant term of the reduction of e^(x/2) by x^2->x+1.
(history; published version)

#11 by Michel Marcus at Tue Jan 18 02:29:30 EST 2022

STATUS

reviewed

approved

#10 by Joerg Arndt at Tue Jan 18 02:27:12 EST 2022

STATUS

proposed

reviewed

#9 by Amiram Eldar at Tue Jan 18 02:08:37 EST 2022

STATUS

editing

proposed

#8 by Amiram Eldar at Tue Jan 18 01:42:48 EST 2022

EXAMPLE

constant=1.151945206759468821277485451505582743212385890...

#7 by Amiram Eldar at Tue Jan 18 01:42:29 EST 2022

MATHEMATICA

RealDigits[u0, 10] (* A193029 *)

u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]

RealDigits[u1, 10] (* A193030 *)

CROSSREFS

Cf. A000045, A193010, A192232, A193030.

#6 by Amiram Eldar at Tue Jan 18 01:42:05 EST 2022

FORMULA

From Amiram Eldar, Jan 18 2022: (Start)

Equals 1 + Sum_{k>=1} Fibonacci(k-1)/(k!*2^k).

Equals exp(1/4)*(cosh(sqrt(5)/4) - sqrt(5)*sinh(sqrt(5)/4)/5). (End)

STATUS

approved

editing

#5 by Russ Cox at Fri Mar 30 18:57:38 EDT 2012

AUTHOR

_Clark Kimberling (ck6(AT)evansville.edu), _, Jul 14 2011

Discussion

Fri Mar 30

18:57

OEIS Server: https://oeis.org/edit/global/285

#4 by T. D. Noe at Thu Jul 14 19:49:44 EDT 2011

STATUS

proposed

approved

#3 by Clark Kimberling at Thu Jul 14 19:42:17 EDT 2011

STATUS

editing

proposed

#2 by Clark Kimberling at Thu Jul 14 17:33:17 EDT 2011

NAME

allocated for Clark KimberlingDecimal expansion of the constant term of the reduction of e^(x/2) by x^2->x+1.

DATA

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

OFFSET

1,3

COMMENTS

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

EXAMPLE

constant=1.151945206759468821277485451505582743212385890...

MATHEMATICA

f[x_] := Exp[x/2]; 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] (* A193029 *)

u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]

RealDigits[u1, 10] (* A193030 *)

CROSSREFS

Cf. A193010, A192232, A193030.

KEYWORD

allocated

nonn,cons

AUTHOR

Clark Kimberling (ck6(AT)evansville.edu), Jul 14 2011

STATUS

approved

editing