A010464 -id:A010464 - cp's OEIS frontend

A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

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

Offset: 1

Rick L. Shepherd, Jun 24 2006

Side length of the largest equilateral triangle that can be inscribed in a unit square (as stated in MathWorld/Weisstein link).

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			1.03527618041008304939559535049619331339627560527972...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Eric Weisstein's World of Mathematics, Equilateral Triangle.
Index entries for algebraic numbers, degree 4.

Cf. A002193, A019679, A019691, A019884, A010464, A092242, A101263.

RealDigits[Sec[15 Degree],10,120][[1]] (* Harvey P. Dale, Jun 03 2015 *)

sqrt(6) - sqrt(2) \\ Charles R Greathouse IV, Aug 27 2017

Equals sec(Pi/12) = sec(A019679) = sqrt(6) - sqrt(2) = A010464 - A002193 = csc(5*Pi/12) = 1/sin(5*Pi/12) = 1/sin(10*A019691) = 1/A019884.
Equals Product_{k >= 1} 1/(1 - 1/(36*(2*k - 1)^2)). - Antonio Graciá Llorente, Mar 20 2024
From Amiram Eldar, Nov 24 2024: (Start)
Equals 2*A101263.
Equals Product_{k>=1} (1 - (-1)^k/A092242(k)). (End)
Smallest positive of the 4 real-valued roots of x^4-16*x^2+16=0. - R. J. Mathar, Aug 31 2025

A187110 Decimal expansion of sqrt(3/8).

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Apart from leading digits, the same as A174925.

Radius of the circumscribed sphere (congruent with vertices) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			sqrt(3/8)=0.61237243569579452454932101867647284799148687016417..

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Michael Penn, Maximize the area of one ellipse!, YouTube video, 2020.
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010464, A010466, A020781, A040001, A040005, A115754, A301755.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron). - Stanislav Sykora, Feb 10 2014

Mathematica
```
RealDigits[N[Sqrt[3/8],300]][[1]]
```

sqrt(3/8) \\ Charles R Greathouse IV, Mar 16 2022

Equals A010464/4. - Stefano Spezia, Jan 26 2025

Equals 3*A020781 = A115754/2 = sqrt(A301755). - Hugo Pfoertner, Jan 26 2025

A020781 Decimal expansion of 1/sqrt(24).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Radius of the inscribed sphere (tangent to the faces) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			1/sqrt(24) = 0.20412414523193150818310700622549094933... . - _Vladimir Joseph Stephan Orlovsky_, May 30 2010

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. Platonic solids inradii: A020763 (octahedron), A179294 (icosahedron), A237603 (dodecahedron). - Stanislav Sykora, Feb 25 2014

Cf. A010464, A010480, A020763, A020853, A021028, A187110, A244980.

RealDigits[N[1/Sqrt[24],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

Equals A010464/12. - Stefano Spezia, Jan 26 2025

Equals 1/A010480 = A020763/2 = 2*A020853 = A187110/3 = A244980/Pi. - Hugo Pfoertner, Jan 26 2025

A010547 Decimal expansion of square root of 96.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 9 followed by {1, 3, 1, 18} repeated. - Harry J. Smith, Jun 11 2009

This differs only by offset from 2*(6^(1/2))/5 = 0.9797958971132712392789... as used in Theorem 5, equation 1.8, p.4 of Cao. - Jonathan Vos Post, Apr 29 2010

			9.797958971132712392789136298823565567863789922626680513730770269003841...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Zhenwei Cao and Alexander Elgart, On efficiency of Hamiltonian--based quantum computation for low-rank matrices, arXiv:1004.4911 [math-ph], 2010-2012.
Index entries for algebraic numbers, degree 2.

Cf. A010167 (continued fraction).

RealDigits[N[96^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 24 2012 *)

{ default(realprecision, 20080); x=sqrt(96); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010547.txt", n, " ", d)); } \\ Harry J. Smith, Jun 11 2009

Equals 4*A010464. - R. J. Mathar, Feb 03 2025

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

A040003 Continued fraction for sqrt(6).

Original entry on oeis.org

2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4

Offset: 0

N. J. A. Sloane

			2.449489742783178098197284074... = 2 + 1/(2 + 1/(4 + 1/(2 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 01 2009

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 143.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010464 (decimal expansion).
Equals twice A040001.
Essentially the same as A010694.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[6], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(6)); for (n=0, 20000, write("b040003.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

a(n-1) = gcd(2^n, 3^n+1) (empirical). - Michel Marcus, Sep 03 2020

G.f.: 2*(1 + x + x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A154235 a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).

Original entry on oeis.org

1, 8, 54, 352, 2276, 14688, 94744, 611072, 3941136, 25418368, 163935584, 1057300992, 6819052096, 43979406848, 283644733824, 1829363802112, 11798463078656, 76094066608128, 490767902078464, 3165202550546432

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(6) = 6.4494897427....
Binomial transform of A164550, second binomial transform of A164549, third binomial transform of A123011, fourth binomial transform of A164532.
Binomial transform is A164551, second binomial transform is A164552, third binomial transform is A164553.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-10).

Cf. A010464 (decimal expansion of square root of 6), A123011, A164532, A164549, A164550, A164551, A164552, A164553.

a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-10*a[n-2]; od; a; # G. C. Greubel, May 21 2019

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

LinearRecurrence[{8, -10}, {1, 8}, 30] (* or *) Table[Simplify[((4 + Sqrt[6])^n -(4-Sqrt[6])^n)/(2*Sqrt[6])], {n, 30}] (* G. C. Greubel, Sep 06 2016 *)

a(n)=([0,1; -10,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

my(x='x+O('x^30)); Vec(x/(1-8*x+10*x^2)) \\ G. C. Greubel, May 21 2019

[lucas_number1(n,8,10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 8*a(n-1) - 10*a(n-2) for n > 1, where a(0)=0, a(1)=1.
G.f.: x/(1 - 8*x + 10*x^2). (End)

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 04 2009

A265300 Decimal expansion of Sum_{k>=1} (x-c(2k-1)), where c = convergents to (x = sqrt(6)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 13 2015

			sum = 0.454587113065072474998978330809543013325085397...

Cf. A010464, A265301, A265302, A265288 (guide).

x = Sqrt[6]; 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]]  (* A265300 *)
RealDigits[s2, 10, 120][[1]]  (* A265301 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265302 *)

A265301 Decimal expansion of Sum_{k>=1} c(2k), where c = convergents to (x = sqrt(6)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 13 2015

			sum = 0.0510257748284475366870974186164101059561600202984037166...

Cf. A010464, A265300, A265302, A265288 (guide).

x = Sqrt[6]; 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]]  (* A265300 *)
RealDigits[s2, 10, 120][[1]]  (* A265301 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265302 *)

A265302 Decimal expansion of Sum_{k>=1} (c(2k)-c(2k-1)), where c = convergents to sqrt(6).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 13 2015

			sum = 0.50561288789352001168607574942595311928124541813374...

Cf. A010464, A265300, A265301, A265288 (guide).

x = Sqrt[6]; 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]]  (* A265300 *)
RealDigits[s2, 10, 120][[1]]  (* A265301 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265302 *)

A154237 a(n) = ( (6 + sqrt(6))^n - (6 - sqrt(6))^n )/(2*sqrt(6)).

Original entry on oeis.org

1, 12, 114, 1008, 8676, 73872, 626184, 5298048, 44791056, 378551232, 3198883104, 27030060288, 228394230336, 1929828955392, 16306120554624, 137778577993728, 1164159319286016, 9836554491620352, 83113874320863744, 702269857101754368

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

nonn

Fifth binomial transform of A002533 without initial term 1. Sixth binomial transform of 1 followed by A056452.

Lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(6) = 8.4494897427....

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12, -30).

Cf. A010464 (decimal expansion of square root of 6), A002533, A056452.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-30*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Sep 07 2016

Join[{a=1,b=12},Table[c=12*b-30*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011*)
LinearRecurrence[{12, -30}, {1, 12}, 25] (* or *) Table[( (6 + sqrt(6))^n - (6 - sqrt(6))^n )/(2*sqrt(6)), {n,1,25}] (* G. C. Greubel, Sep 07 2016 *)

[lucas_number1(n,12,30) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 12*a(n-1) - 30*a(n-2) for n > 1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 12*x + 30*x^2). (End)

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

cp's OEIS Frontend

A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

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A187110 Decimal expansion of sqrt(3/8).

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A020781 Decimal expansion of 1/sqrt(24).

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A010547 Decimal expansion of square root of 96.

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A040003 Continued fraction for sqrt(6).

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A154235 a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).

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