Revision History for A254795
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#8 by Bruno Berselli at Tue Feb 24 05:31:28 EST 2015
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#7 by Joerg Arndt at Tue Feb 24 04:04:53 EST 2015
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#6 by Peter Bala at Mon Feb 23 14:23:05 EST 2015
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#5 by Peter Bala at Mon Feb 23 14:17:29 EST 2015
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| KEYWORD
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nonn,frac,easy,changed
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| STATUS
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proposed
editing
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#4 by Peter Bala at Mon Feb 23 07:00:24 EST 2015
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Discussion
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| Mon Feb 23
| 07:21
| Michel Marcus: keyword frac
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#3 by Peter Bala at Mon Feb 23 06:24:44 EST 2015
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| COMMENTS
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Brouncker gave the generalized continued fraction expansion 4/Pi = 1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... ))). Osler showed that 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) = L^2/Pi, where L is the lemniscate constant A062539. The sequence of convergents to Osler's continued fraction begins [2/1, 9/41, 543/252, 44115/20013, 4410105/2025, ...]. Here we list76, ... ]. The the (unreduced) numerators of thesethe convergents. See A254796are forin A001147, the sequence of denominators. See A254794 for the decimal expansion ofin L^2/PiA024199.
In extending Brouckner's result, Osler showed that 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) = L^2/Pi, where L is the lemniscate constant A062539. The sequence of convergents to Osler's continued fraction begins [2/1, 9/4, 54/25, 441/200, 4410/2025, ...]. Here we list the (unreduced) numerators of these convergents. See A254796 for the sequence of denominators. See A254794 for the decimal expansion of L^2/Pi.
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| MAPLE
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for n from 2 to 18 do a[n] := 4*a[n-1]+(] + (2*n-1)^2*a[n-2] end do:
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| CROSSREFS
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Cf. A008545, A001147, A024199, A142970, A254794, A254796.
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#2 by Peter Bala at Mon Feb 23 05:31:14 EST 2015
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| NAME
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allocatedNumerators of the convergents of the generalized forcontinued Peterfraction Bala2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))).
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| DATA
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2, 9, 54, 441, 4410, 53361, 747054, 12006225, 216112050, 4334247225, 95353438950, 2292816782025, 59613236332650, 1671463434096225, 50143903022886750, 1606276360166472225, 54613396245660055650, 1967688541203928475625, 74772164565749282073750
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| OFFSET
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0,1
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| COMMENTS
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Brouncker gave the generalized continued fraction expansion 4/Pi = 1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... ))). Osler showed that 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) = L^2/Pi, where L is the lemniscate constant A062539. The sequence of convergents to Osler's continued fraction begins [2/1, 9/4, 54/25, 441/200, 4410/2025, ...]. Here we list the (unreduced) numerators of these convergents. See A254796 for the sequence of denominators. See A254794 for the decimal expansion of L^2/Pi.
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| LINKS
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T. J. Osler, <a href="http://www.rowan.edu/colleges/csm/departments/math/facultystaff/osler/130%20Missing%20fractions%20in%20Brouncker.pdf">The missing fractions in Brouncker’s sequence of continued fractions for Pi</a>
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| FORMULA
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a(2*n-1) = ( A008545(n) )^2 = ( Product {k = 0..n-1} 4*k + 3 )^2.
a(2*n) = (4*n + 2)*( A008545(n) )^2 = (4*n + 2)*( Product {k = 0..n-1} 4*k + 3 )^2.
a(n) = 4*a(n-1) + (2*n - 1)^2*a(n-2) with a(0) = 2, a(1) = 9.
a(2*n) = (4*n + 2)*a(2*n-1); a(2*n+1) = (4*n + 4)*a(2*n) + a(2*n-1).
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| MAPLE
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a[0] := 2: a[1] := 9:
for n from 2 to 18 do a[n] := 4*a[n-1]+(2*n-1)^2*a[n-2] end do:
seq(a[n], n = 0 .. 18);
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| CROSSREFS
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Cf. A008545, A142970, A254794, A254796.
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| KEYWORD
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allocated
nonn,easy
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| AUTHOR
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Peter Bala, Feb 23 2015
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| STATUS
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approved
editing
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#1 by Peter Bala at Sun Feb 08 05:28:44 EST 2015
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| NAME
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allocated for Peter Bala
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| KEYWORD
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allocated
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| STATUS
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approved
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