A115754 -id:A115754 - cp's OEIS frontend

A198755 Decimal expansion of x>0 satisfying x^2+cos(x)=2.

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

Offset: 1

Clark Kimberling, Oct 30 2011

For many choices of a,b,c, there is a unique x>0 satisfying a*x^2+b*cos(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c..... x
... 1.... 2..... A198755
... 1.... 3..... A198756
... 1.... 4..... A198757
... 2.... 3..... A198758
... 2.... 4..... A198811
... 3.... 3..... A198812
... 3.... 4..... A198813
... 4.... 3..... A198814
... 4.... 4..... A198815
... 1.... 0..... A125578
.. -1.... 1..... A198816
.. -1.... 2..... A198817
.. -1.... 3..... A198818
.. -1.... 4..... A198819
.. -2.... 1..... A198821
.. -2.... 2..... A198822
.. -2.... 3..... A198823
.. -2.... 4..... A198824
.. -2... -1..... A198825
.. -3.... 0..... A197807
.. -3.... 1..... A198826
.. -3.... 2..... A198828
.. -3.... 3..... A198829
.. -3.... 4..... A198830
.. -3... -1..... A198835
.. -3... -2..... A198836
.. -4.... 0..... A197808
.. -4.... 1..... A198838
.. -4.... 2..... A198839
.. -4.... 3..... A198840
.. -4.... 4..... A198841
.. -4... -1..... A198842
.. -4... -2..... A198843
.. -4... -3..... A198844
... 0.... 1..... A010503
... 0.... 3..... A115754
... 1.... 2..... A198820
... 1.... 3..... A198827
... 1.... 4..... A198837
... 2.... 3..... A198869
... 3.... 4..... A198870
.. -1.... 1..... A198871
.. -1.... 2..... A198872
.. -1.... 3..... A198873
.. -1.... 4..... A198874
.. -2... -1..... A198875
.. -2.... 3..... A198876
.. -3... -2..... A198877
.. -3... -1..... A198878
.. -3.... 1..... A198879
.. -3.... 2..... A198880
.. -3.... 3..... A198881
.. -3.... 4..... A198882
.. -4... -3..... A198883
.. -4... -1..... A198884
.. -4.... 1..... A198885
.. -4.... 3..... A198886
... 0.... 1..... A020760
... 1.... 2..... A198868
... 1.... 3..... A198917
... 1.... 4..... A198918
... 2.... 3..... A198919
... 2.... 4..... A198920
... 3.... 4..... A198921
.. -1.... 1..... A198922
.. -1.... 2..... A198924
.. -1.... 3..... A198925
.. -1.... 4..... A198926
.. -2... -1..... A198927
.. -2.... 1..... A198928
.. -2.... 2..... A198929
.. -2.... 3..... A198930
.. -2.... 4..... A198931
.. -3... -1..... A198932
.. -3.... 1..... A198933
.. -3.... 2..... A198934
.. -3.... 4..... A198935
.. -4... -3..... A198936
.. -4... -2..... A198937
.. -4... -1..... A198938
.. -4.... 1..... A198939
.. -4.... 2..... A198940
.. -4.... 3..... A198941
.. -4.... 4..... A198942
... 1.... 2..... A198923
... 1.... 3..... A198983
... 1.... 4..... A198984
... 2.... 3..... A198985
... 3.... 4..... A198986
.. -1.... 1..... A198987
.. -1.... 2..... A198988
.. -1.... 3..... A198989
.. -1.... 4..... A198990
.. -2... -1..... A198991
.. -2.... 1..... A198992
.. -2... -3..... A198993
.. -3... -2..... A198994
.. -3... -1..... A198995
.. -2.... 1..... A198996
.. -3.... 2..... A198997
.. -3.... 3..... A198998
.. -3.... 4..... A198999
.. -4... -3..... A199000
.. -4... -1..... A199001
.. -4.... 1..... A199002
.. -4.... 3..... A199003
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A198755, take f(x,u,v)=x^2+u*cos(x)-v and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			1.32562251814753662348322902938798744330...

Index entries for transcendental numbers.

Cf. A197737, A198414.

(* Program 1:  A198655 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.32, 1.33}, WorkingPrecision -> 110]
RealDigits[r] (* A198755 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 3}]}, {u, -5, 4}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A198755 *)

A016945 a(n) = 6*n+3.

Original entry on oeis.org

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297, 303, 309, 315, 321, 327

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(37).
Continued fraction expansion of tanh(1/3).
If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
Leaves of the Odd Collatz-Tree: a(n) has no odd predecessors in all '3x+1' trajectories where it occurs: A139391(2*k+1) <> a(n) for all k; A082286(n)=A006370(a(n)). - Reinhard Zumkeller, Apr 17 2008
Let random variable X have a uniform distribution on the interval [0,c] where c is a positive constant. Then, for positive integer n, the coefficient of determination between X and X^n is (6n+3)/(n+2)^2, that is, A016945(n)/A000290(n+2). Note that the result is independent of c. For the derivation of this result, see the link in the Links section below. - Dennis P. Walsh, Aug 20 2013
Positions of 3 in A020639. - Zak Seidov, Apr 29 2015
a(n+2) gives the sum of 6 consecutive terms of A004442 starting with A004442(n). - Wesley Ivan Hurt, Apr 08 2016
Numbers k such that Fibonacci(k) mod 4 = 2. - Bruno Berselli, Oct 17 2017
Also numbers k such that t^k == -1 (mod 7), where t is a member of A047389. - Bruno Berselli, Dec 28 2017

Friedrich L. Bauer, Der (ungerade) Collatz-Baum, Informatik Spektrum 31 (Springer, April 2008), pp. 379-384.
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Dennis P. Walsh, The correlation for a power curve on nonnegative support.
Eric Weisstein's World of Mathematics, Collatz Problem.
Index entries for sequences related to 3x+1 (or Collatz) problem.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Third row of A092260.
Cf. A004442, A008588, A010503, A016921, A016933, A016957, A016969, A019679, A115754.
Subsequence of A061641; complement of A047263; bisection of A047241.
Cf. A000225. - Loren Pearson, Jul 02 2009
Cf. A020639. - Zak Seidov, Apr 29 2015
Odd numbers in A355200.

List([0..60], n-> 3*(1+2*n)); # G. C. Greubel, Sep 18 2019

a016945 = (+ 3) . (* 6)
a016945_list = [3, 9 ..]
-- Wesley Ivan Hurt, Sep 29 2013

[6*n+3 : n in [0..60]]; // Wesley Ivan Hurt, Sep 29 2013

seq(6*n+3, n=0..60); # Dennis P. Walsh, Aug 20 2013
A016945:=n->6*n+3; # Wesley Ivan Hurt, Sep 29 2013

Range[3, 350, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[6n+3, {n, 0, 60}] (* Wesley Ivan Hurt, Sep 29 2013 *)
LinearRecurrence[{2, -1}, {3, 9}, 55] (* Ray Chandler, Jul 17 2015 *)
CoefficientList[Series[3(1+x)/(1-x)^2, {x, 0, 60}], x] (* Robert G. Wilson v, Dec 14 2016 *)

makelist(6*n+3, n, 0, 60); /* Wesley Ivan Hurt, Sep 29 2013 */

{a(n) = 6*n + 3} \\ Wesley Ivan Hurt, Sep 29 2013

x='x+O('x^60); Vec(3*(1+x)/(1-x)^2) \\ Altug Alkan, Apr 08 2016

[3*(1+2*n) for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 3*(2*n + 1) = 3*A005408(n), odd multiples of 3.
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A103333(n+1). - Reinhard Zumkeller, Feb 24 2009
a(n) = 12*n - a(n-1) for n>0, a(0)=3. - Vincenzo Librandi, Nov 20 2010
G.f.: 3*(1+x)/(1-x)^2. - Mario C. Enriquez, Dec 14 2016
E.g.f.: 3*(1 + 2*x)*exp(x). - G. C. Greubel, Sep 18 2019
Sum_{n>=0} (-1)^n/a(n) = Pi/12 (A019679). - Amiram Eldar, Dec 10 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)/2 (A010503).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(3/2) (A115754). (End)
a(n) = (n+2)^2 - (n-1)^2. - Alexander Yutkin, Mar 15 2025

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)

A157697 Decimal expansion of sqrt(2/3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 04 2009

Height (from a vertex to the opposite face) of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
The eccentricity of the ellipse of minimum area that is circumscribing two equal and externally tangent circles (Kotani, 1995). - Amiram Eldar, Mar 06 2022
The standard deviation of a roll of a 3-sided die. - Mohammed Yaseen, Feb 23 2023

			0.81649658092772603273242802490196379732198249355222...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (168) on page 32.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jisho Kotani, Problem 2053, Crux Mathematicorum, Vol. 5, No. 1 (1995), p. 202; Solution to Problem 2053, by David Hankin, ibid., Vol. 22, No. 4 (1996), pp. 187-188.
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985), pp. 449-457.
Index entries for algebraic numbers, degree 2.

Cf. A002193, A020763, A092259, A115754, A145439.

Sqrt(2/3); // G. C. Greubel, Mar 30 2018

Maple
```
evalf(sqrt(2/3)) ;
```

RealDigits[Sqrt[2/3], 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

sqrt(2/3) \\ G. C. Greubel, Mar 30 2018

Equals 1 - (1/2)/2 + (1*3)/(2*4)/2^2 - (1*3*5)/(2*4*6)/2^3 + ... [Jolley]
Equals Sum_{n>=0} (-1)^n*binomial(2n,n)/8^n = 1/A115754. Averaging this constant with sqrt(2) = A002193 = Sum_{n>=0} binomial(2n,n)/8^n yields A145439.
From Michal Paulovic, Dec 08 2022: (Start)
Equals 2 * A020763.
Has periodic continued fraction expansion [0, 1, 4; 2, 4]. (End)
Equals exp(-arctanh(1/5)). - Amiram Eldar, Jul 10 2023
Equals Product_{k>=1} (1 + (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

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

A382713 Simple continued fraction expansion of sqrt(3/2).

Original entry on oeis.org

1, 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, 2

Offset: 0

N. J. A. Sloane, Apr 08 2025

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

Cf. A105397, A115754, A142238, A142239.

Essentially the same as A106469, A040003, A010694.

with(numtheory); cfrac (sqrt(3/2, 70, 'quotients');

PadRight[{1}, 100, {2, 4}] (* Paolo Xausa, Apr 14 2025 *)

def A382713(n): return 1<<1+(n&1) if n else 1 # Chai Wah Wu, Apr 09 2025

A277644 Beatty sequence for sqrt(6)/2.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 84, 85, 86

Offset: 1

Jason Kimberley, Oct 26 2016

Eggleton et al. show that k is in this sequence if and only if A277515(k)=3.

			a(5)=6 because the quotient of 3*5^2 by 2 is 37 which lies between 6^2 and 7^2.

R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Jason Kimberley, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A000196, A032528, A115754, A277515. Complement of A277645.

Magma
```
[Isqrt(3*n^2 div 2): n in [1..60]];
```

Floor[Range[100]*Sqrt[3/2]] (* Paolo Xausa, Jul 11 2024 *)

a(n)=sqrtint(3*n^2\2) \\ Charles R Greathouse IV, Jul 11 2024

a(n) = floor(n*sqrt(6)/2).

a(n) = A000196(A032528(n)).

A379800 Decimal expansion of (1+sqrt(6))/2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jan 19 2025

			1.7247448713915890490986420373529456959829737403283...

Cf. A115754, A176051, A379801.

First[RealDigits[(1 + Sqrt[6])/2, 10, 100]] (* Paolo Xausa, Jan 21 2025 *)

A386739 Decimal expansion of the volume of a sphenocorona with unit edges.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 01 2025

The sphenocorona is Johnson solid J_86.

			1.5153516399764065597284793124718129048228695068...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Sphenocorona.

Cf. A010482 (surface area - 2), A178809 (surface area + 4).

Cf. A010507, A020775, A115754, A386740, A386741.

First[RealDigits[Sqrt[1 + 3*Sqrt[3/2] + Sqrt[13 + Sqrt[54]]]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J86", "Volume"], 10, 100]]

Equals sqrt(1 + 3*sqrt(3/2) + sqrt(13 + 3*sqrt(6)))/2 = sqrt(1 + 3*A115754  + sqrt(13 + A010507))/2.
Equals A386740 - A020775.
Equals the largest real root of 1024*x^8 - 1024*x^6 - 3008*x^4 - 96*x^2 + 9.

cp's OEIS Frontend

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A016945 a(n) = 6*n+3.

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

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