A232328 A generalized Engel expansion of 1/Pi.

4, 3, 6, 12, 51, 146, 280, 482, 687, 3825, 5646, 30904, 120121, 1344923, 2340376, 4456271, 194324055, 219784933, 976224357, 11584437417, 26402463827, 34635051144, 85031207055, 95014277980, 257962314442

Offset: 0

Author

Peter Bala, Nov 27 2013

Keywords

• nonn
• easy

Comments

For a description of two kinds of generalized Engel expansion of a real number see A232327. Compare with A006283 and A014012.

Links

• Eric Weisstein's World of Mathematics, Pierce Expansion
• Wikipedia, Engel Expansion

Crossrefs

Cf. A006283, A014012, A232327.

Programs

Maple

#A232328
#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..24);

Formula

Define the map g(x) 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. Then a(n) = |ceiling(1/g^n(-1/Pi))| for n >= 0.

Generalized Engel series expansion: 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) - - + +.