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A232328
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A generalized Engel expansion of 1/Pi.
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4
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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
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OFFSET
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0,1
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COMMENTS
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For a description of two kinds of generalized Engel expansion of a real number see A232327. Compare with A006283 and A014012.
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LINKS
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FORMULA
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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) - - + +.
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MAPLE
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#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);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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