A001902 -id:A001902 - cp's OEIS frontend

A001900 Successive numerators of Wallis's approximation to Pi/2 (unreduced).

1, 2, 4, 16, 64, 384, 2304, 18432, 147456, 1474560, 14745600, 176947200, 2123366400, 29727129600, 416179814400, 6658877030400, 106542032486400, 1917756584755200, 34519618525593600, 690392370511872000, 13807847410237440000, 303772643025223680000

Offset: 0

N. J. A. Sloane

a(n) = number of permutations of [n+1] all of whose non-initial left-to-right minima are at even positions in the permutation. For example, a(2) = 4 counts 123, 132, 213, 312. - David Callan, Jul 22 2008
Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal. a(2) = 4: [(0,0),(1,0),(2,0)], [(0,0),(0,1),(1,0),(2,0)], [(0,0),(0,1),(0,2),(1,1),(2,0)], [(0,0),(1,0),(0,1),(0,2),(1,1),(2,0)]. - Alois P. Heinz, Mar 23 2017
a(n+1) is the number of 0-1 square matrices of order n+1 with 2n+1 nonzero entries where the cell (i,j) is 1 for all i+j=n+2 and every diagonal, parallel to the main diagonal, has exactly one 1. For example, a(2) = 4: [(0,1,1), (1,1,0), (1,0,0)], [(0,1,1), (0,1,0), (1,1,0)], [(0,0,1), (1,1,1), (1,0,0)], [(0,0,1), (0,1,1), (1,1,0)]. - Christian Barrientos, Jul 17 2021

			From _Wolfdieter Lang_, Dec 06 2017: (Start)
Partial products of the rows N (for numerators a(n)) and D (for denominators b(n) = A000246(n+1)) begin:
n:    0  1  2  3  4   5    6     7      8       9       10 ...
N:    1  2  2  4  4   6    6     8      8      10       10 ...
D:    1  1  3  3  5   5    7     7      9       9       11 ...
a(n): 1  2  4 16 64 384 2304 18432 147456 14745601 4745600 ...
b(n): 1  1  3  9 45 225 1575 11025  99225   893025 9823275 ...  (End)

H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.

Alois P. Heinz, Table of n, a(n) for n = 0..449
John Derbyshire, Prime Obsession, Plume books, p. 16, 2003.
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.
Index to divisibility sequences

Cf. A000246, A284230.

For the reduced form see A001901(n)/A001902(n), n >= 0.

a[n_] := a[n] = If[n==0, 1, (n+Mod[n, 2]) a[n-1]];
a /@ Range[0, 21] (* Jean-François Alcover, Jan 31 2020 *)

a(n)=if(n<0,0,prod(k=1,n,if(k%2,k+1,k)))

2.2.4.4.6.6....2n.2n.../1.3.3.5.5.7.7....(2n-1).(2n+1) ...for n >= 1.
a(n) = 2^n * A010551(n) = 2^n * [n/2]! * [(n+1)/2]!. - Ralf Stephan, Mar 11 2004
Conjecture: a(n) - a(n-1) - n*(n-1)*a(n-2) = 0. - R. J. Mathar, Jun 07 2013 [The proof, for n >= 2, follows from the bisection recurrence given below. - Wolfdieter Lang, Dec 07 2017]
E.g.f.: E(0), where E(k)= 1 + 2*x*(k+1)/((2*k+1) - x*(2*k+1)/(x + 1/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 08 2013
G.f.: G(0), where G(k)= 1 + 2*x*(k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 08 2013
Bisection: a(2*k+1) = ((2*k+1)+1)*a(2*k), a(2*k) = 2*k*a(2*k-1), k >= 0, with a(0) = 1. The proof is obvious from the numbers in the numerator (see the row N in the example). From a proposal by David James Sycamore, Nov 02 2017 based on the fractions 4/1, 8/3, 32/9, 128/45, ... converging very slowly to Pi, given on p. 16 of the Derbyshire link. - Wolfdieter Lang, Dec 06 2017
From Mathias Zechmeister, Jul 24 2022: (Start)
Let J_0(x) and J_1(x) denote the Bessel functions and i = sqrt(-1).
a(n) = denominator([x^n](J_0(x) + J_1(x))).
a(n) = denominator([x^n](J_0(i*x) - i*J_1(i*x))).
Bisection: a(2*k) = A002454(k), a(2*k+1) = A002474(k).
G.f. for 1/a(n): J_0(i*x) - i*J_1(i*x).  (End)

A161736 Denominators of the column sums of the BG2 matrix.

Original entry on oeis.org

1, 9, 75, 1225, 19845, 160083, 1288287, 41409225, 1329696225, 10667118605, 85530896451, 1371086188563, 21972535073125, 176021737014375, 1409850293610375, 90324408810638025, 5786075364399106425, 46326420401234675625, 370882277949065911875, 5938020471163465810125

Offset: 2

Johannes W. Meijer, Jun 18 2009

The BG2 matrix coefficients, see also A008956, are defined by BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .
A different way to define the matrix coefficients is BG2[2*m,n] = (1/m)*sum(LAMBDA(2*m-2*k,n-1)*BG2[2*k,n], k=0..m-1) with LAMBDA(2*m,n-1) = (1-2^(-2*m))*zeta(2*m)-sum((2*k-1)^(-2*m), k=1..n-1) and BG2[0,n] = Pi/2 for m = 0, 1, 2, .. , and n = 1, 2, 3 .. , with zeta(m) the Riemann zeta function.
The columns sums of the BG2 matrix are defined by sb(n) = sum(BG2[2*m,n], m=0..infinity) for n = 2, 3, .. . For large values of n the value of sb(n) approaches Pi/2.
It is remarkable that if we assume that BG2[2m,1] = 2 for m = 0, 1, .. the columns sums of the modified matrix converge to the original sb(n) values. The first Maple program makes use of this phenomenon and links the sb(n) with the central factorial numbers A008956.
The column sums sb(n) can be linked to other sequences, see the second Maple program.
We observe that the column sums sb(n) of the BG2(n) matrix are related to the column sums sl(n) of the LG2(n) matrix, see A008956, by sb(n) = (-1)^(n+1)*(2*n-1)*sl(n).
a(n+2), for n >= 0, seems to coincide with the numerators belonging to A278145. - Wolfdieter Lang, Nov 16 2016
Suppose that, given values f(x-2*n+1), f(x-2*n+3), ..., f(x-1), f(x+1), ..., f(x+2*n-3), f(x+2*n-1), we approximate f(x) using the first 2*n terms of its Taylor series. Then 1/sb(n+1) is the coefficient of f(x-1) and f(x+1). - Matthew House, Dec 03 2024

			sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

G. C. Greubel, Table of n, a(n) for n = 2..830
R. Arratia, S. Garibaldi, and J. Kilian, Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process, arXiv:1310.7055 [math.PR], 2013.

Cf. Numerators A161737.
Cf. A001818, A001902, A008956, A013777, A134372 and A134375.
Cf. A001147, A133221, A161738 and A002474, A278145.

[Denominator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018

nmax := 18; for n from 0 to nmax do A001818(n) := (doublefactorial(2*n-1))^2 od: for n from 0 to nmax do A008956(n, 0):=1 od: for n from 0 to nmax do A008956(n, n) := A001818(n) od: for n from 1 to nmax do for m from 1 to n-1 do A008956(n, m) := (2*n-1)^2*A008956(n-1, m-1) + A008956(n-1, m) od: od: for n from 1 to nmax do for m from 0 to n do s(n, m):=0; s(n, m) := s(n, m)+ sum((-1)^k1*A008956(n, n-k1), k1=0..n-m): od: sb1(n+1) := sum(s(n, k1), k1=1..n) * 2/A001818(n); od: seq(sb1(n), n=2..nmax); # End program 1
nmax1 := nmax; for n from 0 to nmax1 do A001147(n):= doublefactorial(2*n-1) od: for n from 0 to nmax1/2 do A133221(2*n+1) := A001147(n); A133221(2*n) := A001147(n) od: for n from 0 to nmax1 do A002474(n) := 2^(2*n+1)*n!*(n+1)! od: for n from 1 to nmax1 do A161738(n) := ((product((2*n-3-2*k1), k1=0..floor(n/2-1)))) od: for n from 2 to nmax1 do sb2(n) := A002474(n-2) / (A161738(n)*A133221(n-1))^2 od: seq(sb2(n), n=2..nmax1); # End program 2
# Above Maple programs edited by Johannes W. Meijer, Sep 25 2012
r := n -> (1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> numer(simplify(r(n))):
seq(a(n), n = 1..21);  # Peter Luschny, Feb 12 2025

sb[2]=2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2; Table[sb[n] // Denominator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *)

{a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))}; /* Michael Somos, May 09 2011 */

a(n) = denom(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161737(n) = numer(sb(n)).
a(n+1) = numerator of C(2*n,n)^2 * n / 2^(n+1). - Michael Somos, May 09 2011
a(n) = A001902(2*n-3). - Mats Granvik, Nov 25 2018
a(n) = numerator((1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 13 2025

A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).

Original entry on oeis.org

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

N. J. A. Sloane

If p is prime, then a(p-2) == - A001902(p-2) (mod p). Cf. A064169 (third comment) and my formula here. Such pseudoprimes are 1467, 7831, ... Primes p such that a(p-2) == - A001902(p-2) (mod p^2) are 5, 45827, ... Cf. A355959, see also A330719 (third comment). - Thomas Ordowski, Oct 19 2024

			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)

H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.

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

Denominators are A001902. Subsequence of A000079.

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 *)

(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

A069955 Let W(n) = Product_{k=1..n} (1 - 1/4k^2), the partial Wallis product (lim_{n->oo} W(n) = 2/Pi); then a(n) = numerator(W(n)).

Original entry on oeis.org

1, 3, 45, 175, 11025, 43659, 693693, 2760615, 703956825, 2807136475, 44801898141, 178837328943, 11425718238025, 45635265151875, 729232910488125, 2913690606794775, 2980705490751054825, 11912508103174630875, 190453061649520333125, 761284675790187924375

Offset: 0

Benoit Cloitre, Apr 27 2002

Equivalently, denominators in partial products of the following approximation to Pi: Pi = Product_{n>=1} 4*n^2/(4*n^2-1). Numerators are in A056982.

Orin J. Farrell and Bertram Ross, Solved Problems in Analysis, Dover, NY, 1971; p. 77.

Seiichi Manyama, Table of n, a(n) for n = 0..832
Boris Gourévitch, L'univers de Pi.

Not the same as A001902(n).
Cf. A056982 (denominators), A001790, A046161.
W(n)=(3/4)*(A120995(n)/A120994(n)), n>=1.

a[n_] := Numerator[Product[1 - 1/(4*k^2), {k, 1, n}]]; Array[a, 20, 0] (* Amiram Eldar, May 07 2025 *)

a(n) = numerator(prod(k=1, n, 1-1/(4*k^2))); \\ Michel Marcus, Oct 22 2016

a(n) = numerator(W(n)), where W(n) = (2*n)!*(2*n+1)!/((2^n)*n!)^4.
W(n) = (2*n+1)*(binomial(2*n,n)/2^(2*n))^2 = (2*n+1)*(A001790(n)/A046161(n))^2 in lowest terms.
a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n+1/2,n). - Peter Luschny, Apr 08 2016

A092798 Numerator of partial products in an approximation of Pi/2.

Original entry on oeis.org

2, 16, 8192, 274877906944, 5070602400912917605986812821504, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 1

Ralf Stephan, Mar 05 2004

			The first approximations are 2^(1/2), (16/3)^(1/4), (8192/243)^(1/8), (274877906944/215233605)^(1/16).

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270; arXiv:math/0506319 [math.NT], 2005-2006.
J. Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
J. 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.

Denominators are in A092799.

Cf. A000246, A001900, A001901, A001902, A122214, A122216.

for(m=1, 7, p=1; for(n=1, m, p=p*p*(prod(k=1, ceil(n/2), (2*k)^binomial(n, 2*k-1))/(prod(k=1, floor(n/2)+1, (2*k-1)^binomial(n, 2*k-2))))); print1(numerator(p), ", "))

a(n) = Product_{k=1..n+1} A122214(k)^2^(n-k+1). - Jonathan Sondow, Sep 13 2006

a(n) = Numerator(Product_{k=1..n+1} (A122216(k)/A122217(k))^2^(n-k+1)). - Jonathan Sondow, Sep 13 2006

A092799 Denominator of partial products in an approximation to Pi/2.

Original entry on oeis.org

1, 3, 243, 215233605, 2849452841966467687734375, 34139907905802495953388390516678108673704867996275424957275390625

Offset: 1

Ralf Stephan, Mar 05 2004

			The first approximations are 2^(1/2), (16/3)^(1/4), (8192/243)^(1/8), (274877906944/215233605)^(1/16).

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270; arXiv:math/0506319 [math.NT], 2005-2006.
J. Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
J. 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.

Numerators are in A092798.
Cf. A000246, A001900, A001901, A001902.
Cf. A122215, A122217.

for(m=1, 7, p=1; for(n=1, m, p=p*p*(prod(k=1, ceil(n/2), (2*k)^binomial(n, 2*k-1))/(prod(k=1, floor(n/2)+1, (2*k-1)^binomial(n, 2*k-2))))); print1(denominator(p), ", "))

a(n) = Product_{k=1..n+1} A122215(k)^2^(n-k+1). - Jonathan Sondow, Sep 13 2006

a(n) = Denominator(Product_{k=1..n+1} (A122216(k)/A122217(k))^2^(n-k+1)). - Jonathan Sondow, Sep 13 2006

A092053 Denominators of the convergents of the continued fraction expansion [1;1/2,1/3,1/4,...,1/n,...].

Original entry on oeis.org

1, 1, 7, 19, 53, 81, 823, 5359, 12923, 21877, 102061, 354883, 808865, 1433689, 25699639, 369784999, 817787423, 1487830821, 6512750579, 23917578595, 51908057021, 96040578001, 827937066989, 6166467806391, 13211837015707

Offset: 1

Paul D. Hanna, Feb 19 2004

Numerators of convergents are A001902 (successive denominators of Wallis's product approximation to Pi/2). Sum of numerators and denominators equals powers of 2: A001902(n) + a(n) = 2^A092054(n).

Robert Israel, Table of n, a(n) for n = 1..1663

Cf. A001902, A092054.

R:= gfun:-rectoproc({r(n) = (r(n - 1))/(n - 1) + r(n - 2), r(1) = 0, r(2) = 1}, r(n), remember):
seq(numer(R(n)),n=2..30); # Robert Israel, May 14 2017

Numerator[RecurrenceTable[{r[n] == (r[n - 1])/(n - 1) + r[n - 2], r[1] == 0, r[2] == 1}, r, {n, 2, 30}]] (* Terry D. Grant, May 07 2017, fixed by Vaclav Kotesovec, Aug 14 2021 *)
Table[Numerator[ContinuedFractionK[1, 1/k , {k, 1, n}]], {n, 1, 30}] (* Vaclav Kotesovec, Aug 14 2021 *)

a(n)=local(A);CF=contfracpnqn(vector(n,k,1/k));A=denominator(CF[1,1]/CF[2,1])

a(n) = 2^A092054(n) - A001902(n).

A092054 Base-2 logarithm of the sum of numerator and denominator of the convergents of the continued fraction expansion [1; 1/2, 1/3, 1/4, ..., 1/n, ...].

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 11, 14, 15, 16, 18, 20, 21, 22, 26, 30, 31, 32, 34, 36, 37, 38, 41, 44, 45, 46, 48, 50, 51, 52, 57, 62, 63, 64, 66, 68, 69, 70, 73, 76, 77, 78, 80, 82, 83, 84, 88, 92, 93, 94, 96, 98, 99, 100, 103, 106, 107, 108, 110, 112, 113, 114, 120, 126, 127, 128, 130

Offset: 1

Paul D. Hanna, Feb 19 2004

nonn

Consider the convergents of the continued fraction expansion [1; 1/2, 1/3, 1/4, ..., 1/n, ...]. The numerators of the convergents are A001902 (successive denominators of Wallis's product approximation to Pi/2) and the denominators of the convergents are A092053. The sum of the numerators and the denominators equals a power of 2: A001902(n) + A092053(n) = 2^a(n).
Also, a(n-1) is the number of the comparisons that Floyd's heap-construction algorithm will use, in the worst case, to create an n-element heap. See Wikipedia link, section "Building a heap". - Marek A. Suchenek, Mar 16 2014:
First differences appear to be essentially A136480. - Chris Boyd, Jan 14 2016

			a(4)=6 since [1; 1/2, 1/3, 1/4] = 1 + 1/(1/2 + 1/(1/3 + 1/(1/4))) = 45/19; and the sum of the numerator and denominator of 45/19 equals 45 + 19 = 2^6.

Marek A. Suchenek, Elementary Yet Precise Worst-Case Analysis of Floyd's Heap-Construction Program, Fundamenta Informaticae 120 (2012), pp 75--92.
Wikipedia, Binary heap

Cf. A001902, A092053, A136480.

{a(n)=local(A);CF=contfracpnqn(vector(n,k,1/k)); A=length(binary(numerator(1+CF[1,1]/CF[2,1])))-1}

2^a(n) = A001902(n) + A092053(n).
It appears that a(n) = Sum_{k=1..n} A001511(floor((k+1)/2)). Equivalently, a(n) = 2n + 1 - A000120(n) - A000120(n+1) = A011371(n) + A011371(n+1). - Franklin T. Adams-Watters, Feb 02 2006
a(n-1) = 2*n - 2*A000120(n) - A007814(n); see Suchenek link for a proof. - Marek A. Suchenek, Mar 16 2014

cp's OEIS Frontend

A001900 Successive numerators of Wallis's approximation to Pi/2 (unreduced).

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A161736 Denominators of the column sums of the BG2 matrix.

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A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).

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A069955 Let W(n) = Product_{k=1..n} (1 - 1/4k^2), the partial Wallis product (lim_{n->oo} W(n) = 2/Pi); then a(n) = numerator(W(n)).

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A092798 Numerator of partial products in an approximation of Pi/2.

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A092799 Denominator of partial products in an approximation to Pi/2.

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A092053 Denominators of the convergents of the continued fraction expansion [1;1/2,1/3,1/4,...,1/n,...].

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