A002530 -id:A002530 - cp's OEIS frontend

A041111 Denominators of continued fraction convergents to sqrt(63).

1, 1, 15, 16, 239, 255, 3809, 4064, 60705, 64769, 967471, 1032240, 15418831, 16451071, 245733825, 262184896, 3916322369, 4178507265, 62415424079, 66593931344, 994730462895, 1061324394239, 15853271982241

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 14 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,16,0,-1).

Cf. A010516, A041110, A002530.

Denominator/@Convergents[Sqrt[63],30] (* Harvey P. Dale, May 18 2011 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 16 x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 24 2013 *)

From Colin Barker, Jul 15 2012: (Start)
a(n) = 16*a(n-2) - a(n-4).
G.f.: (1+x-x^2)/(1-16*x^2+x^4). (End)
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(14) + sqrt(18) )/2 and beta = ( sqrt(14) - sqrt(18) )/2 be the roots of the equation x^2 - sqrt(14)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even. a(n) = Product_{k = 1..floor((n-1)/2)} ( 14 + 4*cos^2(k*Pi/n) ). Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 14*a(2*n) + a(2*n - 1). (End)

A041039 Denominators of continued fraction convergents to sqrt(24).

Original entry on oeis.org

1, 1, 9, 10, 89, 99, 881, 980, 8721, 9701, 86329, 96030, 854569, 950599, 8459361, 9409960, 83739041, 93149001, 828931049, 922080050, 8205571449, 9127651499, 81226783441, 90354434940, 804062262961

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 8 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A010480, A041038, A002530.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[24],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Denominator[Convergents[Sqrt[24],30]] (* or *) LinearRecurrence[{0,10,0,-1},{1,1,9,10},30] (* Harvey P. Dale, Apr 12 2022 *)

G.f.: (1+x-x^2)/(1-10*x^2+x^4). - Colin Barker, Jan 01 2012
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = sqrt(2) + sqrt(3) and beta = sqrt(2) - sqrt(3) be the roots of the equation x^2 - sqrt(8)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 8 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 8*a(2*n) + a(2*n - 1). (End)

A042937 Denominators of continued fraction convergents to sqrt(1000).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 53, 114, 281, 4329, 8939, 22207, 142181, 164388, 306569, 470957, 777526, 1248483, 78183472, 79431955, 157615427, 237047382, 394662809, 631710191, 4184923955, 9001558101, 22188040157, 341822160456, 705832361069, 1753486882594

Offset: 0

N. J. A. Sloane

			sqrt(1000) = 31.62... = 31 + 1/(1 + 1/(1 + ...)) with convergents 31/1, 32/1, 63/2, 95/3, 158/5, ... - _M. F. Hasler_, Nov 02 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78960998, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A042936 (numerators), A040968 (continued fraction), A010467 (decimals).

Analog for sqrt(m): A000129 (m=2), A002530 (m=3), A001076 (m=5), A041007 (m=6), A041009 (m=7), A041011 (m=8), A005663 (m=10), A041015 (m=11), A041017 (m=12), ..., A042933 (m=998), A042935 (m=999).

Denominator[Convergents[Sqrt[1000], 30]] (* Vincenzo Librandi, Feb 01 2014 *)

A42937=contfracpnqn(c=contfrac(sqrt(1000)),#c-1)[2,] \\ Possibly incorrect last term ignored. NB: a(n) = A42937[n+1]. For more terms use e.g. \p999, or compute any a(n) from this as in A042936. - M. F. Hasler, Nov 01 2019

More terms from Vincenzo Librandi, Feb 01 2014

A143643 Numerators of the lower principal convergents and the lower intermediate convergents to 3^(1/2).

Original entry on oeis.org

1, 3, 5, 12, 19, 45, 71, 168, 265, 627, 989, 2340, 3691, 8733, 13775, 32592, 51409, 121635, 191861, 453948, 716035, 1694157, 2672279, 6322680, 9973081, 23596563, 37220045, 88063572, 138907099, 328657725, 518408351, 1226567328, 1934726305, 4577611587, 7220496869, 17083879020, 26947261171, 63757904493, 100568547815

Offset: 1

Clark Kimberling, Aug 27 2008

The lower principal and intermediate convergents to 3^(1/2), beginning with 1/1, 3/2, 5/3, 12/7, 19/11, form a strictly increasing sequence; with essentially, numerators being this sequence and denominators being A005246.
sqrt(floor(a(n)^2/3)+1) = A005246(n+1). Also see A082630. - Richard R. Forberg, Nov 14 2013
a(n) = U_n(sqrt(6),1) for n odd and a(n) = 3*U_n(sqrt(6),1) for n even, where U_n(sqrt(R),Q) denotes the Lehmer sequence with parameters R and Q. This sequence is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this sequence is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Sep 03 2019

			From _Peter Bala_, Sep 03 2019: (Start)
If p(n)/q(n) denotes the n-th convergent to the simple continued fraction alpha = [c(0); c(1), c(2), ...] then a lower semiconvergent is a rational number of the form ( p(2*n) + m*p(2*n+1) )/( q(2*n) + m*q(2*n+1) ) where 0 <= m <= c(2*n+2). The lower semiconvergents include the even-indexed convergents p(2*n)/q(2*n) and give an increasing sequence of approximations to alpha from below.
In this case the simple continued fraction expansion sqrt(3) = [1; 1, 2, 1, 2, ...] produces the sequence of convergents (p(n)/q(n))n>=0 = [1/1, 2/1, 5/3, 7/4, 19/11, 26,15, 71/41, ...].
Thus the increasing sequence of lower semiconvergents begins 1/1, (1 + 2)/(1 + 1) = 3/2, (1 + 2*2)/(1 + 2*1) = 5/3, (5 + 7)/(3 + 4) = 12/7, (5 + 2*7)/(3 + 2*4) = 19/11, ... with numerators 1, 3, 5, 12, 19, .... (End)

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126.

Wikipedia, Lehmer sequence

Cf. A002530, A002531, A005246, A082630, A143607, A143609, A143642.

a(n) = 4*a(n-2)-a(n-4). G.f.: x*(1+3*x+x^2)/(1-4*x^2+x^4). - Colin Barker, Apr 28 2012

A144535 Numerators of continued fraction convergents to sqrt(3)/2.

Original entry on oeis.org

0, 1, 6, 13, 84, 181, 1170, 2521, 16296, 35113, 226974, 489061, 3161340, 6811741, 44031786, 94875313, 613283664, 1321442641, 8541939510, 18405321661, 118973869476, 256353060613, 1657092233154, 3570537526921, 23080317394680, 49731172316281, 321467351292366

Offset: 0

N. J. A. Sloane, Dec 29 2008

			0, 1, 6/7, 13/15, 84/97, 181/209, 1170/1351, 2521/2911, 16296/18817, 35113/40545, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Cf. A126664, A144536, A002531/A002530.

Bisections give A001570, A011945.

I:=[0, 1, 6, 13]; [n le 4 select I[n] else 14*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

with(numtheory); Digits:=200: cf:=convert(evalf(sqrt(3)/2,confrac); [seq(nthconver(cf,i), i=0..100)];

CoefficientList[Series[x (1 + 6 x - x^2)/((1 - 4 x + x^2) (1 + 4 x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *)
Numerator[Convergents[Sqrt[3]/2,30]] (* or *) LinearRecurrence[{0,14,0,-1},{0,1,6,13},30] (* Harvey P. Dale, Feb 10 2014 *)

Vec(x*(1+6*x-x^2)/((1-4*x+x^2)*(1+4*x+x^2)) + O(x^30)) \\ Colin Barker, Mar 27 2016

From Colin Barker, Apr 14 2012: (Start)
a(n) = 14*a(n-2) - a(n-4).
G.f.: x*(1 + 6*x - x^2)/((1 - 4*x + x^2)*(1 + 4*x + x^2)). (End)
a(n) = ((-(-2-sqrt(3))^n*(-3+sqrt(3)) + (2-sqrt(3))^n*(-3+sqrt(3)) - (3+sqrt(3))*((-2+sqrt(3))^n - (2+sqrt(3))^n)))/(8*sqrt(3)). - Colin Barker, Mar 27 2016
a(2*n) = 6*a(2*n-1) + a(2*n-2). a(2*n+1) = A003154(A101265(n+1)). - John Elias, Dec 10 2021

A144536 Denominators of continued fraction convergents to sqrt(3)/2.

Original entry on oeis.org

1, 1, 7, 15, 97, 209, 1351, 2911, 18817, 40545, 262087, 564719, 3650401, 7865521, 50843527, 109552575, 708158977, 1525870529, 9863382151, 21252634831, 137379191137, 296011017105, 1913445293767, 4122901604639, 26650854921601, 57424611447841, 371198523608647

Offset: 0

N. J. A. Sloane, Dec 29 2008

			0, 1, 6/7, 13/15, 84/97, 181/209, 1170/1351, 2521/2911, 16296/18817, 35113/40545, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
John Elias, Double-Snowflake Fraction
Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Cf. A126664, A144535, A002531/A002530.

Bisections give A011943, A028230.

with(numtheory); Digits:=200: cf:=convert(evalf(sqrt(3)/2,confrac); [seq(nthconver(cf,i), i=0..100)];

Denominator[Convergents [Sqrt[3]/2, 30]] (* Vincenzo Librandi, Feb 01 2014 *)
LinearRecurrence[{0,14,0,-1},{1,1,7,15},30] (* Harvey P. Dale, Sep 15 2017 *)

G.f.: (1 + x - 7*x^2 + x^3)/(1 - 14*x^2 + x^4). - Colin Barker, Jan 01 2012
a(n) = 14*a(n-2) - a(n-4). - Sergei N. Gladkovskii, Jun 07 2015
a(n) = ((3+sqrt(3))*((-2+sqrt(3))^n + (2+sqrt(3))^n) - (-3+sqrt(3))*((-2-sqrt(3))^n + (2-sqrt(3))^n))/12. - Vaclav Kotesovec, Jun 08 2015
From John Elias, Dec 02 2021: (Start)
a(2*n) = 6*A001353(n)^2 + 1. See illustration in links.
a(2*n+1) = 2*a(2*n) + a(2*n-1). (End)

A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).

Original entry on oeis.org

8, 0, 2, 5, 8, 3, 0, 9, 0, 8, 0, 3, 5, 1, 4, 8, 3, 4, 3, 7, 7, 8, 7, 4, 1, 8, 1, 2, 6, 3, 0, 4, 2, 4, 9, 6, 0, 5, 8, 6, 0, 4, 7, 7, 6, 8, 9, 5, 1, 3, 2, 7, 7, 7, 0, 1, 7, 7, 4, 1, 8, 5, 1, 4, 8, 4, 1, 4, 0, 6, 0, 8, 4, 4, 8, 7, 0, 0, 3, 0, 2, 2, 1, 7, 9, 4

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.8025830908035148343778741812630...

Cf. A002194, A002530, A002531, A265295, A265296, A265288 (guide).

x := sqrt(3) - 2:
evalf(2*sqrt(3)*add( x^(n*(n+1)/2)/(x^n - 1), n = 1..18), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

From Peter Bala, Aug 24 2022: (Start)
Equals 2*sqrt(3)*Sum_{n >= 1} 1/( 1 + (2+sqrt(3))^(2*n-1) ).
A more rapidly converging series for the constant is 2*sqrt(3)*Sum_{n >= 1} x^(n*(n+1)/2)/(x^n - 1), where x = sqrt(3) - 2. See A001227. (End)

A265295 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(3).

Original entry on oeis.org

2, 8, 7, 2, 8, 0, 0, 8, 0, 0, 8, 3, 4, 8, 8, 3, 9, 3, 5, 1, 1, 4, 5, 1, 5, 3, 9, 8, 7, 6, 6, 8, 3, 3, 1, 6, 8, 2, 3, 9, 0, 9, 4, 2, 0, 8, 6, 4, 5, 6, 7, 1, 8, 7, 9, 3, 8, 7, 1, 6, 8, 2, 6, 8, 1, 3, 8, 8, 3, 8, 6, 4, 1, 0, 7, 1, 6, 8, 0, 0, 6, 4, 0, 8, 2, 6

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.28728008008348839351145153987668331682390...

Cf. A002194, A002530, A002531, A265294, A265296, A265288 (guide).

x := 7 - 4*sqrt(3):
evalf(2*sqrt(3)*add( x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..10), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^n)/(1 - x^n), where x = 7 - 4*sqrt(3). - Peter Bala, Aug 24 2022

A265296 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(3).

Original entry on oeis.org

1, 0, 8, 9, 8, 6, 3, 1, 7, 0, 8, 8, 7, 0, 0, 3, 2, 2, 7, 8, 8, 9, 3, 2, 5, 7, 2, 1, 1, 3, 9, 7, 2, 5, 8, 1, 2, 8, 8, 2, 5, 1, 4, 1, 9, 7, 7, 5, 9, 6, 9, 9, 9, 6, 4, 9, 5, 6, 4, 5, 8, 6, 7, 8, 2, 9, 8, 0, 2, 4, 4, 7, 2, 5, 5, 5, 8, 6, 8, 3, 0, 8, 6, 2, 6, 2

Offset: 1

Clark Kimberling, Dec 07 2015

			sum = 1.0898631708870032278893257211397258128825141977596999...

Cf. A002194, A002530, A002531, A265294, A265295, A265288 (guide).

x := 2 - sqrt(3):
evalf(2*sqrt(3)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..13), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), where x = 2 - sqrt(3). - Peter Bala, Aug 24 2022

A321119 a(n) = ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2); n-th row common denominator of A321118.

Original entry on oeis.org

4, 2, 8, 10, 28, 38, 104, 142, 388, 530, 1448, 1978, 5404, 7382, 20168, 27550, 75268, 102818, 280904, 383722, 1048348, 1432070, 3912488, 5344558, 14601604, 19946162, 54493928, 74440090, 203374108, 277814198, 759002504, 1036816702, 2832635908, 3869452610

Offset: 0

Franck Maminirina Ramaharo, Nov 01 2018

			a(0) = ((1 - sqrt(3))^0 + (1 + sqrt(3))^0)/2^floor((0 - 1)/2) = 2*(1 + 1) = 4.

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.2.

Encyclopedia of Mathematics, Quadrature formula
John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.
Peter Köhler, On the weights of Sard's quadrature formulas, CALCOLO Vol. 25 (1988), 169-186.
Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.
Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.
Isaac J. Schoenberg, Spline interpolation and best quadrature formulae, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A002176 (common denominators of Cotesian numbers).
Cf. A321118, A321120.
Cf. A001834, A002530, A002531, A003500, A005246, A048788, A083336, A125905.

LinearRecurrence[{0, 4, 0, -1}, {4, 2, 8, 10}, 50]

a(n) := ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2)$
makelist(ratsimp(a(n)), n, 0, 50);

a(n) = (((sqrt(2) - sqrt(6))/2)^n + ((sqrt(6) + sqrt(2))/2)^n)*((2 - sqrt(2))*(-1)^n + 2 + sqrt(2))/2.
a(-n) = (-1)^n*a(n).
a(n) = 2*A000034(n+1)*A002531(n).
a(2*n) = 2*A001834(n).
a(2*n+1) = 2*A003500(n).
a(n) = 4*a(n-2) - a(n-4) with a(0) = 4, a(1) = 2, a(2) = 8, a(3) = 10.
a(2*n+3) = a(2*n+1) + a(2*n+2).
a(2*n+2) = a(2*n) + 2*a(2*n+1).
G.f.: 2*(1 - x)*(2 + 3*x - x^2)/(1 - 4*x^2 + x^4).
E.g.f.: (1 + exp(-sqrt(6)*x))*((2 - sqrt(2))*exp(sqrt(2 - sqrt(3))*x) + (2 + sqrt(2))*exp(sqrt(2 + sqrt(3))*x))/2.
Lim_{n->infinity} a(2*n+1)/a(2*n) = (1 + sqrt(3))/2.

cp's OEIS Frontend

A041111 Denominators of continued fraction convergents to sqrt(63).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A041039 Denominators of continued fraction convergents to sqrt(24).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A042937 Denominators of continued fraction convergents to sqrt(1000).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A143643 Numerators of the lower principal convergents and the lower intermediate convergents to 3^(1/2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A144535 Numerators of continued fraction convergents to sqrt(3)/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A144536 Denominators of continued fraction convergents to sqrt(3)/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A265295 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(3).

Views

Author

Keywords

A265296 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(3).