A382713 -id:A382713 - cp's OEIS frontend

A115754 Decimal expansion of sqrt(3/2).

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

Offset: 1

Eric Desbiaux, Jul 30 2008

Coordinate of a control point for a degree-5 integration formula for 7 points over the unit circle. [Stroud & Secrest]
Also real and imaginary part of sqrt(-3i). - Alonso del Arte, Dec 11 2012
Area of the quadrilateral obtained when slicing a unit cube with a plane passing through opposite vertices and the middle of opposite edges. See CNRS link. - Michel Marcus, Mar 26 2016
Positive zero of the Hermite polynomial of degree 3. - A.H.M. Smeets, Jun 02 2025

			1.2247448713915890490986420373529456959829737403283350642163...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Ana Rechtman, Mars 2016, 3e défi, Images des Mathématiques, CNRS, 2016.
A. H. Stroud and D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Index entries for algebraic numbers, degree 2

Cf. A382713 (continued fraction), A068388 (Engel expansion).

Cf. A010464 (double), A187110 (half), A157697 (reciprocal).

RealDigits[Sqrt[3/2], 10, 105][[1]] (* Alonso del Arte, Dec 11 2012 *)

Equals 2*A187110.
Equals Sum_{k>=0} binomial(1/2, k)/2^k. - Bruno Berselli, Sep 11 2015
From Amiram Eldar, Aug 02 2020: (Start)
Equals Product_{k>=0} (1 + (-1)^k/(6*k + 3)).
Equals Sum_{k>=0} binomial(2*k,k)/12^k.
Equals 1 + Sum_{k>=1} (2*k - 1)!!/((2*k)!! * 3^k). (End)
Equals A010464/2. - R. J. Mathar, Feb 23 2021

A142238 Numerators of continued fraction convergents to sqrt(3/2).

Original entry on oeis.org

1, 5, 11, 49, 109, 485, 1079, 4801, 10681, 47525, 105731, 470449, 1046629, 4656965, 10360559, 46099201, 102558961, 456335045, 1015229051, 4517251249, 10049731549, 44716177445, 99482086439, 442644523201, 984771132841, 4381729054565, 9748229241971

Offset: 0

N. J. A. Sloane, Oct 05 2008, following a suggestion from Rob Miller (rmiller(AT)AmtechSoftware.net)

From Charlie Marion, Jan 07 2009: (Start)
In general, denominators, a(k,n) and numerators, b(k,n), of continued
fraction convergents to sqrt((k+1)/k) may be found as follows:
a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2)
and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);
b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n) = 2*b(k,2n-1)+b(k,2n-2)
and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).
For example, the convergents to sqrt(3/2) start 1/1, 5/4, 11/9,
49/40, 109/89.
In general, if a(k,n) and b(k,n) are the denominators and numerators,
respectively, of continued fraction convergents to sqrt((k+1)/k)
as defined above, then
k*a(k,2n)^2-a(k,2n-1)*a(k,2n+1)=k=k*a(k,2n-2)*a(k,2n)-a(k,2n-1)^2 and
b(k,2n-1)*b(k,2n+1)-k*b(k,2n)^2=k+1=b(k,2n-1)^2-k*b(k,2n-2)*b(k,2n);
for example, if k=2 and n=3, then b(2,n)=a(n) and
2*a(2,6)^2-a(2,5)*a(2,7)=2*881^2-396*3920=2;
2*a(2,4)*a(2,6)-a(2,5)^2=2*89*881-396^2=2;
b(2,5)*b(2,7)-2*b(2,6)^2=485*4801-2*1079^2=3;
b(2,5)^2-2*b(2,4)*b(2,6)=485^2-2*109*1079=3.
Cf. A000129, A001333, A142239, A153315-A153318. (End)

			The initial convergents are 1, 5/4, 11/9, 49/40, 109/89, 485/396, 1079/881, 4801/3920, 10681/8721, 47525/38804, 105731/86329, ...

Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A115754, A142239, A382713.

with(numtheory): cf := cfrac (sqrt(3)/sqrt(2),100): [seq(nthnumer(cf,i), i=0..50)]; [seq(nthdenom(cf,i), i=0..50)]; [seq(nthconver(cf,i), i=0..50)];

Numerator[Convergents[Sqrt[3/2], 30]] (* Bruno Berselli, Nov 11 2013 *)
LinearRecurrence[{0,10,0,-1},{1,5,11,49},30] (* Harvey P. Dale, Dec 30 2017 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,10,0]^n*[1;5;11;49])[1,1] \\ Charles R Greathouse IV, Jun 21 2015

G.f.'s for numerators and denominators are -(1+5*x+x^2-x^3)/(-1-x^4+10*x^2) and -(1+4*x-x^2)/(-1-x^4+10*x^2).

a(2n) = A041006(2n)/2 = A054320(n), a(2n-1) = A041006(2n-1) = A041038(2n-1) = A001079(n). - M. F. Hasler, Feb 14 2009

A142239 Denominators of continued fraction convergents to sqrt(3/2).

Original entry on oeis.org

1, 4, 9, 40, 89, 396, 881, 3920, 8721, 38804, 86329, 384120, 854569, 3802396, 8459361, 37639840, 83739041, 372596004, 828931049, 3688320200, 8205571449, 36510605996, 81226783441, 361417739760, 804062262961, 3577666791604, 7959395846169, 35415250176280

Offset: 0

N. J. A. Sloane, Oct 05 2008, following a suggestion from Rob Miller (rmiller(AT)AmtechSoftware.net)

sqrt(3/2) = 1.224744871... = 2/2 + 2/9 + 2/(9*89) + 2/(89*881) + 2/(881*8721) + 2/(8721*86329) + ... - Gary W. Adamson, Oct 08 2008
From Charlie Marion, Jan 07 2009: (Start)
In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:
a(k,0) = 1, a(k,1) = 2k;
for n > 0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2) and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);
b(k,0) = 1, b(k,1) = 2k+1;
for n > 0, b(k,2n) = 2*b(k,2n-1)+b(k,2n-2) and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).
For example, the convergents to sqrt(3/2) start 1/1, 5/4, 11/9, 49/40, 109/89.
In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then
k*a(k,2n)^2 - a(k,2n-1)*a(k,2n+1) = k = k*a(k,2n-2)*a(k,2n) - a(k,2n-1)^2 and
b(k,2n-1)*b(k,2n+1) - k*b(k,2n)^2 = k+1 = b(k,2n-1)^2 - k*b(k,2n-2)*b(k,2n);
for example, if k=2 and n=3, then a(2,n)=a(n) and
2*a(2,6)^2 - a(2,5)*a(2,7) = 2*881^2 - 396*3920 = 2;
2*a(2,4)*a(2,6) - a(2,5)^2 = 2*89*881 - 396^2 = 2;
b(2,5)*b(2,7) - 2*b(2,6)^2 = 485*4801 - 2*1079^2 = 3;
b(2,5)^2 - 2*b(2,4)*b(2,6) = 485^2 - 2*109*1079 = 3.
(End)
For n > 0, a(n) equals the permanent of the n X n tridiagonal matrix with the main diagonal alternating sequence [4, 2, 4, 2, 4, ...] and 1's along the superdiagonal and the subdiagonal. - Rogério Serôdio, Apr 01 2018

			The initial convergents are 1, 5/4, 11/9, 49/40, 109/89, 485/396, 1079/881, 4801/3920, 10681/8721, 47525/38804, 105731/86329, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A115754, A142238.

Cf. A000129, A001333, A142238, A153315, A153316, A153317, A153318, A382713.

I:=[1,4,9,40]; [n le 4 select I[n] else 10*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 01 2014

with(numtheory): cf := cfrac (sqrt(3)/sqrt(2),100): [seq(nthnumer(cf,i), i=0..50)]; [seq(nthdenom(cf,i), i=0..50)]; [seq(nthconver(cf,i), i=0..50)];

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[3/2], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[3/2], 30]] (* Bruno Berselli, Nov 11 2013 *)

G.f.'s for numerators and denominators are -(1+5*x+x^2-x^3)/(-1-x^4+10*x^2) and -(1+4*x-x^2)/(-1-x^4+10*x^2).
a(n) = 10*a(n-2) - a(n-4) for n > 3. - Vincenzo Librandi, Feb 01 2014
From: Rogério Serôdio, Apr 02 2018: (Start)
Recurrence formula: a(n) = (3-(-1)^n)*a(n-1) + a(n-2), a(0) = 1, a(1) = 4;
Some properties:
(1) a(n)^2 - a(n-2)^2 = (3-(-1)^n)*a(2*n-1), for n > 1;
(2) a(2*n+1) = a(n)*(a(n+1) + a(n-1)), for n > 0;
(3) a(2*n) = A041007(2*n);
(4) a(2*n+1) = 2*A041007(2*n+1). (End)

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A115754 Decimal expansion of sqrt(3/2).

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A142238 Numerators of continued fraction convergents to sqrt(3/2).

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A142239 Denominators of continued fraction convergents to sqrt(3/2).

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A068388 Engel expansion of sqrt(3/2).

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