A124647 -id:A124647 - cp's OEIS frontend

A093766 Decimal expansion of Pi/(2*sqrt(3)).

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

Offset: 0

Eric W. Weisstein, Apr 15 2004

Density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
The number gives the areal coverage (90.68... percent) of the close hexagonal (densest) packing of circles in the plane. The hexagonal unit cell is a rhombus of side length 1 and height sqrt(3)/2; the area of the unit cell is sqrt(3)/2 and the four parts of circles add to an area of one circle of radius 1/2, which is Pi/4. - R. J. Mathar, Nov 22 2011
Ratio of surface area of a sphere to the regular octahedron whose edge equals the diameter of the sphere. - Omar E. Pol, Dec 09 2013

			0.906899682117108925297039128821077866142033124046370287784942...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 506.
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.
Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 30.

J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discr. Comp. Geom., 13 (1995), 383-403.
Xi Lin, Dirk Schmelter, Sadaf Imanian, and Horst Hintze-Bruening, Hierarchically Ordered alpha-Zirconium Phosphate Platelets in Aqueous Phase with Empty Liquid, Scientific Reports (2019) Vol. 9, Article No. 16389.
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015. See Table 22 for L(m=6,r=2,s=1).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 625-626.
N. J. A. Sloane and Andrey Zabolotskiy, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural).
Eckard Specht, May 21 2012, The best known packings of equal circles in a circle (complete up to N=1500).
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See p. 146.
Eric Weisstein's World of Mathematics, Smoothed Octagon.
Eric Weisstein's World of Mathematics, Circle Packing.
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)*A093602, A346585, A346584, A346583.

Cf. A072691, A381671.

RealDigits[Pi/(2 Sqrt[3]), 10, 111][[1]] (* Robert G. Wilson v, Nov 07 2012 *)

Pi/sqrt(12) \\ Charles R Greathouse IV, Oct 31 2014

Equals (5/6)*(7/6)*(11/12)*(13/12)*(17/18)*(19/18)*(23/24)*(29/30)*(31/30)*..., where the numerators are primes > 3 and the denominators are the nearest multiples of 6.
Equals Sum_{n>=1} 1/A134667(n). [Jolley]
Equals Sum_{n>=0} (-1)^n/A124647(n). [Jolley eq. 273]
Equals A000796 / A010469. - Omar E. Pol, Dec 09 2013
Continued fraction expansion: 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). See A254381 for a sketch proof. - Peter Bala, Feb 04 2015
From Peter Bala, Feb 16 2015: (Start)
Equals 4*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 5)).
Continued fraction: 1/(1 + 1^2/(4 + 5^2/(2 + 7^2/(4 + 11^2/(2 + ... + (6*n + 1)^2/(4 + (6*n + 5)^2/(2 + ... ))))))). (End)
The inverse is (2*sqrt(3))/Pi = Product_{n >= 1} 1 + (1 - 1/(4*n))/(4*n*(9*n^2 - 9*n + 2)) = (35/32) * (1287/1280) * (8075/8064) * (5635/5632) * (72819/72800) * ... =  1.102657790843585... - Dimitris Valianatos, Aug 31 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 3) dx.
Equals Integral_{x=0..oo} 1/(3*x^2 + 1) dx. (End)
Equals 1 + Sum_{k>=1} ( 1/(6*k+1) - 1/(6*k-1) ). - Sean A. Irvine, Jul 24 2021
For positive integer k, Pi/(2*sqrt(3)) = Sum_{n >= 0} (6*k + 4)/((6*n + 1)*(6*n + 6*k + 5)) - Sum_{n = 0..k-1} 1/(6*n + 5). - Peter Bala, Jul 10 2024
From Stefano Spezia, Jun 05 2025: (Start)
Equals Sum_{k>=0} (-1)^k/((k + 1)*(3*k + 1)).
Equals Integral_{x=0..oo} 1/(x^4 + x^2 + 1) dx.
Equals Integral_{x=0..oo} x^2/(x^4 + x^2 + 1) dx. (End)
Equals sqrt(A072691) = 3*A381671. - Hugo Pfoertner, Jun 05 2025

Entry revised by N. J. A. Sloane, Feb 10 2013

A165665 a(n) = (32^n - 2) 2^n.

Original entry on oeis.org

1, 8, 40, 176, 736, 3008, 12160, 48896, 196096, 785408, 3143680, 12578816, 50323456, 201310208, 805273600, 3221159936, 12884770816, 51539345408, 206157905920, 824632672256, 3298532786176, 13194135339008, 52776549744640

Offset: 0

Klaus Brockhaus, Sep 24 2009

Binomial transform of A058481. Second binomial transform of (A082505 without initial term 0). Third binomial transform of A010686.
Partial sums are in A060867.
a(n) is the sum of the odd numbers taken progressively by moving through them by 2^n-tuples. a(0)=1; a(1) = 3+5=8; a(2) = 7+9+11+13 = 40; a(3) = 15+17+19+21+23+25+27+29 = 176; a(n) = sum_{k=0,1,..,A000225(n)} (A000225(n+1)+2*k).  - J. M. Bergot, Dec 06 2014
The number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A058481, A082505, A010686 (repeat 1, 5), A060867, A010036, A124647.

Magma
```
[ (3*2^n-2)*2^n: n in [0..23] ];
```

Table[(3*2^n-2)2^n,{n,0,30}] (* or  *) LinearRecurrence[{6,-8},{1,8},30] (* Harvey P. Dale, Nov 18 2020 *)

a(n)=(3*2^n-2)*2^n \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 6*a(n-1)-8*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
a(n) = 8*A010036(n-1) for n > 0.
G.f.: (2*x+1)/((1-2*x)*(1-4*x)).
E.g.f.: 3*e^(4*x) - 2*e^(2*x). - Robert Israel, Dec 15 2014

A171220 a(n) = (2n + 1)*5^n.

Original entry on oeis.org

1, 15, 125, 875, 5625, 34375, 203125, 1171875, 6640625, 37109375, 205078125, 1123046875, 6103515625, 32958984375, 177001953125, 946044921875, 5035400390625, 26702880859375, 141143798828125, 743865966796875, 3910064697265625, 20503997802734375, 107288360595703125

Offset: 0

Jaume Oliver Lafont, Dec 05 2009

Inserting x=1/sqrt(b) into the power series expansion of arctanh(x) yields the general BBP-type formula log((sqrt(b)+1)/(sqrt(b)-1))*sqrt(b)/2 = Sum_{k>=0} 1/((2k+1)b^k).
This sequence illustrates case b=5, with
Sum_{k>=0} 1/a(k) = sqrt(5)*log((1+sqrt(5))/2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David H. Bailey, Compendium of BBP formulas for mathematical constants. See formula 86 at p. 26.
Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A014480 ((2n+1)*2^n), A124647 ((2n+1)*3^n), A058962 ((2n+1)*4^n), A155988 ((2n+1)*9^n), A165283 ((2n+1)*16^n), A166725 ((2n+1)*25^n).

[(2*n+1)*5^n: n in [0..25]]; // Vincenzo Librandi, Jun 08 2011

PARI
```
a(n)=(2*n+1)*5^n
```

a(n) = 10*a(n-1) - 25*a(n-2).
O.g.f: (1+5*x)/(1-5*x)^2.
Sum_{n>=0} (-1)^n/a(n) = sqrt(5)*arctan(1/sqrt(5)). - Amiram Eldar, Feb 26 2022
E.g.f.: exp(5*x)*(1 + 10*x). - Stefano Spezia, May 09 2023

A199299 a(n) = (2n + 1)6^n.

Original entry on oeis.org

1, 18, 180, 1512, 11664, 85536, 606528, 4199040, 28553472, 191476224, 1269789696, 8344332288, 54419558400, 352638738432, 2272560758784, 14575734521856, 93096626946048, 592433080565760, 3757718396731392, 23765029860409344, 149902496042582016, 943288877536247808

Offset: 0

Philippe Deléham, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A014480, A058962, A124647, A155988, A171220, A199300, A199301.

[(2*n+1)*6^n: n in [0..30]]; // Vincenzo Librandi, Nov 05 2011

a[n_] := (2*n + 1)*6^n; Array[a, 25, 0] (* Amiram Eldar, Dec 10 2022 *)

a(n) = (2*n+1)*6^n \\ Amiram Eldar, Dec 10 2022

a(n) = 12*a(n-1) - 36*a(n-2).
G.f.: (1+6*x)/(1-6*x)^2.
a(n) = 6*a(n-1) + 2*6^n. - Vincenzo Librandi, Nov 05 2011
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(6)*arccoth(sqrt(6)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(6)*arccot(sqrt(6)). (End)
E.g.f.: exp(6*x)*(1 + 12*x). - Stefano Spezia, May 07 2023

A199300 a(n) = (2n + 1)7^n.

Original entry on oeis.org

1, 21, 245, 2401, 21609, 184877, 1529437, 12353145, 98001617, 766718533, 5931980229, 45478515089, 346032180025, 2616003280989, 19668469112621, 147174406808233, 1096686708796833, 8142067989552245, 60251303122686613, 444556912229552577, 3271482918202092041

Offset: 0

Philippe Deléham, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A014480, A058962, A124647, A155988, A171220, A199299, A199301.

[(2*n+1)*7^n: n in [0..30]]; // Vincenzo Librandi, Nov 05 2011

a[n_] := (2*n + 1)*7^n; Array[a, 25, 0] (* Amiram Eldar, Dec 10 2022 *)
LinearRecurrence[{14,-49},{1,21},30] (* Harvey P. Dale, Mar 26 2025 *)

a(n) = (2*n+1)*7^n \\ Amiram Eldar, Dec 10 2022

a(n) = 14*a(n-1) - 49*a(n-2).
G.f.: (1+7*x)/(1-7*x)^2.
a(n) = 7*a(n-1) + 2*7^n. - Vincenzo Librandi, Nov 05 2011
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(7)*arccoth(sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(7)*arccot(sqrt(7)). (End)
E.g.f.: exp(7*x)*(1 + 14*x). - Stefano Spezia, May 09 2023

a(15) corrected by Vincenzo Librandi, Nov 05 2011

A199301 a(n) = (2n+1)*8^n.

Original entry on oeis.org

1, 24, 320, 3584, 36864, 360448, 3407872, 31457280, 285212672, 2550136832, 22548578304, 197568495616, 1717986918400, 14843406974976, 127543348822016, 1090715534753792, 9288674231451648, 78812993478983680, 666532744850833408, 5620492334958379008, 47269781688880726016

Offset: 0

Philippe Deléham, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018 (Powers of 8), A005408 (2n+1).

Cf. A014480, A058962, A124647, A155988, A171220, A199299, A199300.

[(2*n+1)*8^n: n in [0..30]]; // Vincenzo Librandi, Nov 05 2011

A199301:=n->(2*n+1)*8^n: seq(A199301(n), n=0..20); # Wesley Ivan Hurt, Oct 30 2014

Table[(2 n + 1)*8^n, {n, 0, 20}] (* Wesley Ivan Hurt, Oct 30 2014 *)

a(n) = (2*n+1)*8^n \\ Amiram Eldar, Dec 10 2022

a(n) = 16*a(n-1)-64*a(n-2).
G.f.: (1+8*x)/(1-8*x)^2.
a(n) = 8*(a(n-1)+2^(3*n-2)). - Vincenzo Librandi, Nov 05 2011
a(n) = A005408(n) * A001018(n). - Wesley Ivan Hurt, Oct 30 2014
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(8)*arccoth(sqrt(8)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(8)*arccot(sqrt(8)). (End)
E.g.f.: exp(8*x)*(1 + 16*x). - Stefano Spezia, May 09 2023

a(18) corrected by Vincenzo Librandi, Nov 05 2011

A340183 a(n) = Product_{1<=j,k,m<=n-1} (4sin(jPi/(2n))^2 + 4sin(kPi/(2n))^2 + 4sin(mPi/(2*n))^2).

Original entry on oeis.org

1, 6, 1157625, 170875128460147163136, 448524809573174705684873233798538664686384705625

Offset: 1

Seiichi Manyama, Dec 31 2020

nonn

(a(n)/(n*3^(n-1)))^(1/3) is an integer.

Cf. A007341, A124647, A340181, A340182.

Round[Table[2^((n-1)^3)* Product[3 - Cos[j*Pi/n] - Cos[k*Pi/n] - Cos[m*Pi/n], {j, 1, n-1}, {k, 1, n-1}, {m, 1, n-1}], {n, 1, 5}]] (* Vaclav Kotesovec, Jan 04 2021 *)

default(realprecision, 500);
{a(n) = round(prod(j=1, n-1, prod(k=1, n-1, prod(m=1, n-1, 4*sin(j*Pi/(2*n))^2+4*sin(k*Pi/(2*n))^2+4*sin(m*Pi/(2*n))^2))))}

a(n) = Product_{1<=i,j,k<=n-1} (4*f(i*Pi/(2*n))^2 + 4*g(j*Pi/(2*n))^2 + 4*h(k*Pi/(2*n))^2), where f(x), g(x) and h(x) are sin(x) or cos(x).

Limit_{n->infinity} a(n)^(1/n^3) = exp(8*A340322/Pi^3). - Vaclav Kotesovec, Jan 05 2021

A340181 a(n) = Product_{1<=j,k,m<=n} (4sin(jPi/(2n+1))^2 + 4sin(kPi/(2n+1))^2 + 4sin(mPi/(2*n+1))^2).

Original entry on oeis.org

1, 9, 7486875, 14334918272193811385583, 1483160703050490588200236172057973908184332257091136

Offset: 0

Seiichi Manyama, Dec 31 2020

nonn

(a(n)/((2n + 1)*3^n))^(1/3) is an integer.

Cf. A124647, A127605, A340182, A340183.

Round[Table[2^(n^3)* Product[3 - Cos[2*j*Pi/(2*n + 1)] - Cos[2*k*Pi/(2*n + 1)] - Cos[2*m*Pi/(2*n + 1)], {j, 1, n}, {k, 1, n}, {m, 1, n}], {n, 0, 5}]] (* Vaclav Kotesovec, Jan 04 2021 *)

default(realprecision, 500);
{a(n) = round(prod(j=1, n, prod(k=1, n, prod(m=1, n, 4*sin(j*Pi/(2*n+1))^2+4*sin(k*Pi/(2*n+1))^2+4*sin(m*Pi/(2*n+1))^2))))}

Limit_{n->infinity} a(n)^(1/n^3) = exp(8*A340322/Pi^3). - Vaclav Kotesovec, Jan 05 2021

A362885 Array read by ascending antidiagonals: A(n, k) = (1 + 2n)k^n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 5, 6, 1, 0, 7, 20, 9, 1, 0, 9, 56, 45, 12, 1, 0, 11, 144, 189, 80, 15, 1, 0, 13, 352, 729, 448, 125, 18, 1, 0, 15, 832, 2673, 2304, 875, 180, 21, 1, 0, 17, 1920, 9477, 11264, 5625, 1512, 245, 24, 1, 0, 19, 4352, 32805, 53248, 34375, 11664, 2401, 320, 27, 1

Offset: 0

Stefano Spezia, May 08 2023

			The array begins:
    1,  1,   1,    1,     1,     1, ...
    0,  3,   6,    9,    12,    15, ...
    0,  5,  20,   45,    80,   125, ...
    0,  7,  56,  189,   448,   875, ...
    0,  9, 144,  729,  2304,  5625, ...
    0, 11, 352, 2673, 11264, 34375, ...
    ...

Cf. A000007 (k=0), A000012 (n=0), A004248, A005408 (k=1), A008585 (n=1), A014480 (k=2), A033429 (n=2), A058962 (k=4), A124647 (k=3), A155988 (k=9), A171220 (k=5), A176043, A199299 (k=6), A199300 (k=7), A199301 (k=8), A244727 (n=3), A362886 (antidiagonal sums).

A[n_,k_]:=(1+2n)k^n; Join[{1}, Table[A[n-k,k],{n,10},{k,0,n}]]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1+k*x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+2k*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n, k) = A005408(n)*A004248(n, k).
O.g.f. of column k: (1 + k*x)/(1 - k*x)^2.
E.g.f. of column k: exp(k*x)*(1 + 2*k*x).
A(n, n) = A176043(n+1).

cp's OEIS Frontend

A093766 Decimal expansion of Pi/(2*sqrt(3)).

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A165665 a(n) = (3*2^n - 2) * 2^n.

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Magma

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A171220 a(n) = (2n + 1)*5^n.

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Magma

PARI

Formula

A199299 a(n) = (2*n + 1)*6^n.

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Magma

Mathematica

PARI

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A199300 a(n) = (2*n + 1)*7^n.

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Magma

Mathematica

PARI

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A199301 a(n) = (2n+1)*8^n.

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Magma

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A340183 a(n) = Product_{1<=j,k,m<=n-1} (4*sin(j*Pi/(2*n))^2 + 4*sin(k*Pi/(2*n))^2 + 4*sin(m*Pi/(2*n))^2).

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A165665 a(n) = (32^n - 2) 2^n.

A199299 a(n) = (2n + 1)6^n.

A199300 a(n) = (2n + 1)7^n.

A340183 a(n) = Product_{1<=j,k,m<=n-1} (4sin(jPi/(2n))^2 + 4sin(kPi/(2n))^2 + 4sin(mPi/(2*n))^2).

A340181 a(n) = Product_{1<=j,k,m<=n} (4sin(jPi/(2n+1))^2 + 4sin(kPi/(2n+1))^2 + 4sin(mPi/(2*n+1))^2).

A362885 Array read by ascending antidiagonals: A(n, k) = (1 + 2n)k^n.