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A232327 A generalized Engel expansion of 1/Pi. 4
3, 23, 27, 89, 137, 9190, 25731, 80457, 125859, 270815, 609977, 959612, 1034186, 1491489, 2975032, 264484387, 1092196976, 1194228023, 1424193547, 4523998315, 13583506006, 380693793416, 1097951708621, 1486580651232 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The Engel expansion of 1/Pi is given in A014012 and the Pierce (or alternating Engel) expansion of 1/Pi is found in A006283.

We can unify the algorithms for finding the Engel and Pierce expansions of a real number as follows.

Define the map f:[-1,1]\{0} -> (-1/2,1) by f(x) = x*ceiling(1/x) - 1 and let f^(n)(x) denote the n-th iterate of f, with the convention that f^(0)(x) = x. Let r be a real number such that 0 < r < 1.

Then the sequence of positive integers e(n) := ceiling(1/f^(n)(r)) is the Engel expansion of r. The associated Engel series representation is r = 1/e(0) + 1/(e(0)*e(1)) + 1/(e(0)*e(1)*e(2)) + ....

The sequence of positive integers p(n) := |ceiling(1/f^(n)(-r))| is the Pierce expansion of r. The associated Pierce series representation is r = 1/p(0) - 1/(p(0)*p(1)) + 1/(p(0)*p(1)*p(2)) - ....

By working with a modification of the map f we can find two generalized Engel-type expansions for the real number r (still assuming 0 < r < 1). To this end, we define the map g:[-1,1]\{0} -> (-1/2,1) by g(x) = -x*ceiling(-1/x) - 1 and let g^(n)(x) denote the n-th iterate of g, with the convention that g^(0)(x) = x.

A)

Our first generalized expansion of r is the integer sequence a(n) := |ceiling(1/g^(n)(r))| for n >= 0 and until g^n(r) = 0. It can be shown that we have a generalized Engel-type representation for r by means of the (possibly infinite) series r = 1/a(0) - 1/(a(0)*a(1)) - 1/(a(0)*a(1)*a(2)) + 1/(a(0)*a(1)*a(2)*a(3)) + 1/(a(0)*a(1)*a(2)*a(3)*a(4)) - - + + ..., where the pattern of signs of the terms is as indicated.

The series will be finite if and only if r is rational.

The present sequence is an example of this first type of generalized Engel expansion for the real number r := 1/Pi.

B)

The second generalized Engel expansion of r is defined as the sequence of integers b(n) := |ceiling(1/g^(n)(-r))| for n >= 0 and until g^(n)(-r) = 0.

It can be shown that we now have a generalized Engel-type representation for r of the form r = 1/b(0) + 1/(b(0)*b(1)) - 1/(b(0)*b(1)*b(2)) - 1/(b(0)*b(1)*b(2)*b(3)) + + - - ....

Again, the series terminates when r is rational, otherwise it is infinite.

See A232328 for the generalized Engel expansion of 1/Pi of the second kind.

We can generalize the Engel and Pierce expansions of a real number even further by considering series expansions to a base b. See A232325 for a definition and details. The usual Engel and Pierce expansions occur when the base b = 1 and the two generalized Engel expansions described above arise when the base b = -1.

LINKS

Table of n, a(n) for n=0..23.

Eric Weisstein's World of Mathematics, Pierce Expansion

Wikipedia, Engel Expansion

FORMULA

a(n) = ceiling(1/g^(n)(1/Pi)), where g(x) = -x*ceiling(-1/x) - 1.

Generalized Engel series expansion:

1/Pi = 1/3 - 1/(3*23) - 1/(3*23*27) + 1/(3*23*27*89) + 1/(3*23*27*89*137) - - + +.

EXAMPLE

Comparison of the Engel, alternating Engel and generalized Engel series expansions for 1/Pi.

A014012: Engel series expansion

1/Pi = 1/4 + 1/(4*4) + 1/(4*4*11) + 1/(4*4*11*45) + 1/(4*4*11*45*70) + ...

A006283: Alternating Engel series expansion

1/Pi = 1/3 - 1/(3*22) + 1/(3*22*118) - 1/(3*22*118*383) + 1/(3*22*118*83*571) - ...

A232327: Generalized Engel series expansion of the first kind

1/Pi = 1/3 - 1/(3*23) - 1/(3*23*27) + 1/(3*23*27*89) + 1/(3*23*27*89*137) - - + + ....

A232328: Generalized Engel series expansion of the second kind

1/Pi = 1/4 + 1/(4*3) - 1/(4*3*6) - 1/(4*3*6*12) + 1/(4*3*6*12*51) + 1/(4*3*6*12*51*146) - - + + ...

MAPLE

#A232327

#Define the n-th iterate of the map f(x) = x/b*ceiling(b/x) - 1

map_iterate := proc(n, b, x) option remember;

if n = 0 then

   x

else

   -1 + 1/b*thisproc(n-1, b, x)*ceil(b/thisproc(n-1, b, x))

end if

end proc:

#Define the terms of the expansion of x to the base b

a := n -> ceil(evalf(b/map_iterate(n, b, x))):

Digits := 500:

#Choose values for x and b

x := 1/Pi: b:= -1:

seq(abs(a(n)), n = 0..25);

CROSSREFS

Cf. A006283, A014012, A232325, A232328.

Sequence in context: A153834 A323010 A153707 * A153708 A109218 A103361

Adjacent sequences:  A232324 A232325 A232326 * A232328 A232329 A232330

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Nov 27 2013

STATUS

approved

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Last modified July 28 11:14 EDT 2021. Contains 346326 sequences. (Running on oeis4.)