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A232328 A generalized Engel expansion of 1/Pi.

%I A232328 #10 Feb 16 2025 08:33:20
%S A232328 4,3,6,12,51,146,280,482,687,3825,5646,30904,120121,1344923,2340376,
%T A232328 4456271,194324055,219784933,976224357,11584437417,26402463827,
%U A232328 34635051144,85031207055,95014277980,257962314442
%N A232328 A generalized Engel expansion of 1/Pi.
%C A232328 For a description of two kinds of generalized Engel expansion of a real number see A232327. Compare with A006283 and A014012.
%H A232328 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>
%H A232328 Wikipedia, <a href="http://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%F A232328 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.
%F A232328 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) - - + +.
%p A232328 #A232328
%p A232328 #Define the n-th iterate of the map f(x) = x/b*ceiling(b/x) - 1
%p A232328 map_iterate := proc(n,b,x) option remember;
%p A232328 if n = 0 then
%p A232328    x
%p A232328 else
%p A232328    -1 + 1/b*thisproc(n-1,b,x)*ceil(b/thisproc(n-1,b,x))
%p A232328 end if
%p A232328 end proc:
%p A232328 #Define the terms of the expansion of x to the base b
%p A232328 a := n -> ceil(evalf(b/map_iterate(n,b,x))):
%p A232328 Digits:= 500:
%p A232328 #Choose values for x and b
%p A232328 x := -1/Pi: b:= -1:
%p A232328 seq(abs(a(n)), n = 0..24);
%Y A232328 Cf. A006283, A014012, A232327.
%K A232328 nonn,easy
%O A232328 0,1
%A A232328 _Peter Bala_, Nov 27 2013