A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).
1, 2, 4, 16, 64, 128, 256, 2048, 16384, 32768, 65536, 262144, 1048576, 2097152, 4194304, 67108864, 1073741824, 2147483648, 4294967296, 17179869184, 68719476736, 137438953472, 274877906944, 2199023255552
Offset: 0
Examples
From _Wolfdieter Lang_, Dec 07 2017: (Start) The Wallis numerators (N) and denominators (D) with partial products A(n) = A001900(n) and B(n) = A000246(n+1) in unreduced form, and a(n) and b(n) = A001902(n) in reduced form. n, k: 0 1 2 3 4 5 6 7 8 9 10 ... N(k): 1 2 2 4 4 6 6 8 8 10 10 ... D(k): 1 1 3 3 5 5 7 7 9 9 9 ... A(n): 1 2 4 16 64 384 2304 18432 147456 1474560 14745600 ... B(n): 1 1 3 9 45 225 1575 11025 99225 893025 9823275 ... a(n): 1 2 4 16 64 128 256 2048 16384 32768 65536 ... b(n): 1 1 3 9 45 75 175 1225 11025 19845 43659 ... n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = numerator(384/225) = numerator(128/75) = 128. (End)
References
- H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.
Links
- Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
- Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
- Eric Weisstein's World of Mathematics, Pi
- Eric Weisstein's World of Mathematics, Pi Continued Fraction
- Index to divisibility sequences
Programs
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Mathematica
a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := ((n - 1)!!^2*(n + 1))/n!!^2; Table[a[n] // Numerator, {n, 0, 23}] (* Jean-François Alcover, Jun 19 2013 *)
Formula
(2*2*4*4*6*6*8*8*...*2n*2n*...)/(1*3*3*5*5*7*7*9*...*(2n-1)*(2n+1)*...) for n >= 1.
From Wolfdieter Lang, Dec 07 2017: (Start)
1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the numerators with offset 0.
a(n) = numerator(W(n)), for n >= 0, with W(n) = Product_{k=0..n} N(k)/D(k) (reduced), with N(k) = 2*floor((k+1)/2) for k >= 1 and N(0) = 1, and D(k) = 2*floor(k/2) + 1, for k >= 0. (End)
a(n) is the numerator of the continued fraction [1;1,1/2,1/3,...,1/n]. - Thomas Ordowski, Oct 19 2024
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