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A340738 Denominator of a sequence of fractions converging to e. 3
1, 2, 7, 18, 71, 252, 1001, 4540, 18089, 99990, 398959, 2602278, 10391023, 78132152, 312129649, 2658297528, 10622799089, 101072656170, 403978495031, 4247085597370, 16977719590391, 195445764537012, 781379079653017, 9775727355457908, 39085931702241241, 528050767520083262, 2111421691000680031 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence is a subset of the numerators of a sequence of fractions converging to e which was obtained by the use of a program which searched for a fraction having a closer value to e than the preceding one. The initial terms of this sequence were 3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540. The subset of the denominators filtered out of this sequence are a(1)..a(8).

The convergence is conjectured.

LINKS

Table of n, a(n) for n=1..27.

FORMULA

a(1) = 1, a(2) = 2; for n > 2, a(n) = (n+2)*a(n-1)/2 - a(n-2) - (n-2)*a(n-3)/2 if n is even, 2*a(n-1) + n*a(n-2) otherwise.

EXAMPLE

Sequence of fractions begins 3/1, 5/2, 19/7, 49/18, 193/71, 685/252, 2721/1001, 12341/4540, ...

MAPLE

e:=proc(a, b, n)option remember; e(a, b, 1):=a; e(a, b, 2):=b; if n>2 and n mod 2 =1 then 2*e(a, b, n-1)+n*e(a, b, n-2) else if n>3 and n mod 2 = 0 then (n+2)*e(a, b, n-1)/2 -(e(a, b, n-2)+(n-2)*e(a, b, n-3)/2) fi fi end

seq(e(1, 2, n), n = 1..20)

# code to print the sequence of fractions and error

for n from 1` to 20 do print(e(3, 5, n)/e(1, 2, n), evalf(exp(1)-e(3, 5, n)/e(1, 2, n)) od

MATHEMATICA

a[1] = 1; a[2] = 2; a[n_] := a[n] = If[EvenQ[n], (n + 2)*a[n - 1]/2 - (a[n - 2] + (n - 2)*a[n - 3]/2), 2*a[n - 1] + n*a[n - 2]]; Array[a, 20] (* Amiram Eldar, Jan 18 2021 *)

CROSSREFS

Numerators are listed in A340737.

Cf. A007676/A007677.

Sequence in context: A185308 A002214 A303742 * A218684 A337614 A343908

Adjacent sequences:  A340735 A340736 A340737 * A340739 A340740 A340741

KEYWORD

nonn,frac

AUTHOR

Gary Detlefs, Jan 18 2021

STATUS

approved

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Last modified June 19 18:35 EDT 2021. Contains 345144 sequences. (Running on oeis4.)