A042944 -id:A042944 - cp's OEIS frontend

A042945 Numbers congruent to 2,3,6,11 mod 12 missing from A042944 (conjectured to be finite).

78, 159, 207, 243, 246, 342, 435, 603, 711, 834, 1422, 1923, 2010, 2022, 2175, 2319, 2454, 2718, 2766, 3150, 3402, 3510, 3711, 3774, 4167, 4854, 4959, 4995, 5283, 6015, 6018, 6666, 6879, 7470, 7863, 10095, 10638, 10923, 11295, 12063, 12534, 13154

Offset: 1

Allan Wilks and Brian Galebach

nonn

Michel Marcus, Table of n, a(n) for n = 1..53
J. Bourgain and A. Kontorovich, On the Strong Density Conjecture for Integral Apollonian Circle Packings, arXiv:1205.4416 [math.NT], 2012.
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Number Theory, arXiv:math/0009113 [math.NT], 2000-2003; J. Number Theory, 100 (2003), 1-45.
Eric Weisstein's World of Mathematics, Bowl of Integers

Cf. A042944, A042946.

A042946 Frequency of occurrence of numbers appearing in A042944.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 6, 8, 4, 4, 4, 4, 4, 4, 8, 8, 6, 8, 4, 4, 4, 8, 8, 8, 4, 8, 12, 2, 8, 8, 12, 8, 8, 4, 4, 4, 4, 12, 8, 4, 4, 10, 8, 8, 8, 4, 4, 4, 12, 8, 8, 8, 12, 12, 8, 8, 8, 12, 6, 8, 12, 8, 4, 12, 8, 12, 12, 8, 12, 16, 4, 4, 8, 12, 8, 6, 8, 12, 12, 4, 4, 8, 12, 8, 4

Offset: 0

Allan Wilks

nonn

			E.g. 35 occurs 6 times.

A294510 Residues modulo 24 of curvatures in the Apollonian circle packing A042944.

Original entry on oeis.org

2, 3, 6, 11, 14, 15, 18, 23

Offset: 1

Jonathan Sondow, Nov 16 2017

Fuchs and Sanden proved that all curvatures in the Apollonian circle packing -1, 2, 3, 6 are congruent mod 24 to either 2, 3, 6, 11, 14, 15, 18, or 23.

Additional comments and links are in A189226.

E. Fuchs and K. Sanden, Some experiments with integral Apollonian circle packings, arXiv:1001.1406 [math.NT], 2010; Experiment. Math., 20 (2011), 380-399.

Cf. A042944, A042945, A189226, A218155.

A248938 Decimal expansion of beta = G^2(2/3)Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)) (where G is Catalan's constant), a constant related to the problem of integral Apollonian circle packings.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 17 2014

			0.4612609086138613...

Steven R. Finch, Apollonian circles with integer curvatures, p. 6. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 256.
Elena Fuchs and Katherine Sanden, Some experiments with integral Apollonian circle packings, arXiv:1001.1406 [math.NT] p. 7.
Peter Sarnak, Integral Apollonian Packings, Princeton MAA Lecture - January, 2009, p. 21.

Cf. A006752, A042944, A042946, A052483, A189226, A189227, A248930.

kmax = 25; Clear[P]; Do[P[k] = Product[p = Prime[n]; If[Mod[p, 4] == 3 , 1 - 2/(p*(p - 1)^2) // N[#, 40]&, 1], {n, 1, 2^k}]; Print["P(", k, ") = ", P[k]], {k, 10, kmax}]; beta = Catalan^2*(2/3)*P[kmax]; RealDigits[beta, 10, 16] // First
(* -------------------------------------------------------------------------- *)
$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (1 - 2/(p*(p - 1)^2));
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*P[4, 3, m]; sump = sump + difp; m++];
RealDigits[Chop[N[2*Catalan^2/3 * Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

beta = (G^2/3)*A248930, where G is Catalan's constant A006752.

More digits from Vaclav Kotesovec, Jun 27 2020

A189226 Curvatures in the nickel-dime-quarter Apollonian circle packing, ordered first by generation and then by size.

Original entry on oeis.org

-11, 21, 24, 28, 40, 52, 61, 157, 76, 85, 96, 117, 120, 132, 181, 213, 237, 376, 388, 397, 132, 156, 160, 189, 204, 205, 216, 237, 253, 285, 288, 309, 316, 336, 349, 405, 412, 421, 453, 460, 469, 472, 517, 544, 565, 616, 628, 685, 717, 741, 1084, 1093, 1104, 1125, 1128, 1140

Offset: 1

Jonathan Sondow, Apr 18 2011

For a circle, curvature = 1/radius. The curvatures of a quarter, nickel, and dime are approximately proportional to 21, 24, and 28, respectively. Three mutually tangent circles with curvatures 21, 24, 28 can be inscribed in a circle of curvature 11.
Apollonius's and Descartes's Theorems say that, given three mutually tangent circles of curvatures a, b, c, there are exactly two circles tangent to all three, and their curvatures are a + b + c +- 2*sqrt(ab + ac + bc). (Here negative curvature of one of the two circles means that the three circles are inscribed in it.)
Fuchs (2009) says "An Apollonian circle packing ... is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well." That is because if a + b + c - 2s*qrt(ab + ac + bc) is an integer, then so is a + b + c + 2*sqrt(ab + ac + bc).
For n > 1, the n-th generation of the packing has 4*3^(n-2) circles.
Infinitely many of the curvatures are prime numbers A189227. In fact, in any integral Apollonian circle packing that is primitive (i.e., the curvatures have no common factor), the prime curvatures constitute a positive fraction of all primes (Bourgain 2012) and there are infinitely many pairs of tangent circles both of whose curvatures are prime (Sarnak 2007, 2011).
Fuchs and Sanden (2012) report on experiments with the nickel-dime-quarter Apollonian circle packing, which they call the coins packing P_C.

			The 1st-generation curvatures are -11, 21, 24, 28, the 2nd are 40, 52, 61, 157, and the 3rd are 76, 85, 96, 117, 120, 132, 181, 213, 237, 376, 388, 397. The 4th generation begins 132, 156, 160, 189, 204, 205, 216, ....
As 21 + 24 + 28 +- 2*sqrt(21*24 + 21*28 + 24*28) = 157 or -11, the sequence begins -11, 21, 24, 28, ... and 157 is in it.
The primes 157 and 397 are the curvatures of two circles that are tangent.

D. Austin, When Kissing Involves Trigonometry, AMS feature column March 2006.
J. Bourgain, Integral Apollonian circle packings and prime curvatures, arXiv:1105.5127 [math.NT], 2011-2012.
J. Bourgain and A. Kontorovich, On the Strong Density Conjecture for Integral Apollonian Circle Packings, arXiv:1205.4416 [math.NT], 2012-2013. See figure 1.
S. Butler, R. Graham, G. Guettler and C. Mallows, Irreducible Apollonian configurations and packings, Discrete & Computational Geometry, 44 (2010), 487-507.
E. Fuchs, Arithmetic Properties of Apollonian Circle Packings, Ph.D. thesis 2009.
E. Fuchs and K. Sanden, Some experiments with integral Apollonian circle packings, Experiment. Math. 20 (2011), 380-399.
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group., Discrete & Computational Geometry, 34 (2005), no. 4, 547-585.
K. E. Hirst, The Apollonian Packing of Circles, J. London Math. Soc. s1-42(1) (1967), 281-291.
E. Kasner and F. Supnick, The Apollonian packing of circles, Proc. Nat. Acad. Sci. U.S.A. 29 (1943), 378-384.
A. Kontorovich, From Apollonius to Zaremba: Local-global phenomena in thin orbits, Bull. Amer. Math. Soc., 50 (2013), 187-228.
J. C. Lagarias, C. L. Mallows, and Allan Wilks, Beyond the Descartes Circle Theorem, Amer. Math Monthly, 109 (2002), 338-361.
L. Levine, W. Pegden, C. K. Smart, Apollonian Structure in the Abelian Sandpile, arXiv:1208.4839 [math.AP], 2012-2014.
D. Mackenzie, A Tisket, a Tasket, an Apollonian Gasket, American Scientist, 98 (2010).
C. L. Mallows, Growing Apollonian Packings, J. Integer Sequences, 12 (2009), article 09.2.1.
I. Peterson, Circle game, Science News, 4/21/01.
I. Peterson, Temple circles, Math Trek, 4/23/01.
P. Sarnak, Letter to Lagarias on integral Apollonian packings, June, 2007.
P. Sarnak, Integral Apollonian packings, MAA Lecture, Jan 2009.
P. Sarnak, Integral Apollonian packings, Amer. Math. Monthly, 118 (2011), 291-306.
K. E. Stange, The sensual Apollonian circle packing, arXiv:1208.4836 [math.NT], 2012-2014.
Wikipedia, Integral Apollonian circle packings
Wikipedia, Descartes' theorem

Cf. A042944, A042946, A042945, A045506, A045673, A045864, A052483, A060790, A135849, A137246, A154636, A154637, A154638, A171090, A189227, A218155, A248938, A265495, A294510.

root = {-11, 21, 24, 28};
triples = Subsets[root, {3}];
a = {root};
Do[
  ng = Table[Total@t + 2 Sqrt@Total[Times @@@ Subsets[t, {2}]], {t, triples}];
  AppendTo[a, Sort@ng];
  triples = Join @@ Table[{t, r} = tr; Table[Append[p, r], {p, Subsets[t, {2}]}], {tr, Transpose@{triples, ng}}]
  , {k, 3}];
Flatten@a (* Andrey Zabolotskiy, May 29 2022 *)

a(n) == 0, 4, 12, 13, 16, or 21 (mod 24).

Terms a(28) and beyond from Andrey Zabolotskiy, May 29 2022

A045506 Inscribe 2 spheres of curvature 2 inside sphere of curvature -1, continue to inscribe spheres where possible; sequence gives list of curvatures.

Original entry on oeis.org

-1, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104

Offset: 0

Brian Galebach

A 3-dimensional Apollonian (or Soddy) problem.

Appears to be congruent to 0 or 2 mod 3.

Cf. A042944, A007494.

Appears to coincide with a(n)=2+3*floor((n-1)/2)+(1+(-1)^n)/2-0^n, in which case the g.f. is (-1+3x+2x^2-x^3)/((1+x)(1-x)^2). - Paul Barry, Sep 11 2008

A060790 Inscribe two circles of curvature 2 inside a circle of curvature -1. Sequence gives curvatures of the smallest circles that can be sequentially inscribed in such a diagram.

Original entry on oeis.org

-1, 2, 2, 3, 15, 38, 110, 323, 927, 2682, 7754, 22403, 64751, 187134, 540822, 1563011, 4517183, 13054898, 37729362, 109039875, 315131087, 910745750, 2632104062, 7606921923, 21984412383, 63536130986, 183622826522, 530679817859, 1533693138351, 4432455434478

Offset: 0

Brian Galebach, Apr 26 2001

The ratio of successive terms approaches the constant phi+sqrt(phi) ~= 2.89005363826396..., where phi is the golden ratio (sqrt(5)+1)/2. The ratio between the curvatures of two successively smaller circles approaches this constant in any apollonian packing as the curvatures increase.

For more comments, references and links, see A189226.

			After circles of 2, 2, 3, 15 have been inscribed in the diagram, the next smallest circle that can be inscribed has a curvature of 38.

Harry J. Smith, Table of n, a(n) for n = 0..200
Jeffrey C. Lagarias, Colin L. Mallows and Allan Wilks, Beyond the Descartes Circle Theorem, arXiv:math/0101066 [math.MG], 2001.
Jeffrey C. Lagarias, Colin L. Mallows and Allan Wilks, Beyond the Descartes Circle Theorem, Amer. Math Monthly, 109 (2002), 338-361.
I. Peterson, Circle Game, Science News, 4/21/01.
Index entries for linear recurrences with constant coefficients, signature (2, 2, 2, -1).

Cf. A042944, A189226, A189227.

a:=[-1,2,2,3];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

seq(coeff(series((x-1)*(1-3*x-3*x^2)/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

CoefficientList[Series[(-3 z^3 + 4 z - 1)/(z^4 - 2 z^3 - 2 z^2 - 2 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{2, 2, 2, -1}, {-1, 2, 2, 3}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

{ for (n=0, 200, if (n>3, a=2*a1 + 2*a2 + 2*a3 - a4; a4=a3; a3=a2; a2=a1; a1=a, if (n==0, a=a4=-1, if (n==1, a=a3=2, if (n==2, a=a2=2, a=a1=3)))); write("b060790.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 12 2009

a(n) = 2a(n-1) + 2a(n-2) + 2a(n-3) - a(n-4).
G.f.: -(1-x)*(1 - 3*x - 3*x^2)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Apr 22 2012
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

Corrected by T. D. Noe, Nov 08 2006

A218155 Numbers congruent to 2, 3, 6, 11 mod 12.

Original entry on oeis.org

2, 3, 6, 11, 14, 15, 18, 23, 26, 27, 30, 35, 38, 39, 42, 47, 50, 51, 54, 59, 62, 63, 66, 71, 74, 75, 78, 83, 86, 87, 90, 95, 98, 99, 102, 107, 110, 111, 114, 119, 122, 123, 126, 131, 134, 135, 138, 143, 146, 147, 150, 155, 158, 159, 162, 167, 170, 171, 174

Offset: 1

Jean-Claude Babois, Oct 22 2012

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A042944, A042945.

LinearRecurrence[{2, -2, 2, -1}, {2, 3, 6, 11}, 100] (* T. D. Noe, Nov 11 2012 *)

for(m=2,175,if(binomial(m,4)%binomial(m,2)==0,print1(m,", "))) \\ Hugo Pfoertner, Aug 11 2020

a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) - a(n-4). - Charles R Greathouse IV, Nov 09 2012
G.f.: x^2*(x^3+4*x^2-x+2) / ((x-1)^2*(x^2+1)). - Colin Barker, Jan 07 2013
{m>1|C(m,4)==0 (mod C(m,2))}. - Gary Detlefs, Jan 11 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(3)+1)*Pi/24 - log(2+sqrt(3))/(4*sqrt(3)) - log(2)/6. - Amiram Eldar, Mar 18 2022

Edited by Andrey Zabolotskiy, Aug 11 2020

A045673 Curvatures in diagram constructed by inscribing 2 circles of curvature 0 and 1 inside circle of curvature 0, continuing indefinitely to inscribe circles wherever possible.

Original entry on oeis.org

0, 1, 4, 9, 12, 16, 24, 25, 28, 33, 36, 40, 49, 52, 57, 60, 64, 72, 73, 76, 81, 84, 88, 96, 97, 100, 105, 108, 112, 121, 124, 129, 136, 144, 145, 148, 153, 156, 160, 168, 169, 172, 177, 180, 184, 192, 193, 196, 201, 204, 216, 217, 220, 225, 228, 232

Offset: 0

Allan Wilks

nonn

Appears to be a superset of {A008784 - 1}. - Ralf Stephan, Jan 26 2005

See A189226 for additional comments, references, links, examples, and crossrefs. - Jonathan Sondow, Aug 24 2012

K. E. Stange, The sensual Apollonian circle packing, arXiv:1208.4836 [math.NT], 2012-2014; see section 14. - _Jonathan Sondow_, Aug 24 2012

Cf. A042944-A042946, A045679, A189226.

A045679 Numbers congruent to 0,1,4,9 mod 12 missing from A045673 (conjectured to be finite).

Original entry on oeis.org

13, 21, 37, 45, 48, 61, 69, 85, 93, 109, 117, 120, 132, 133, 141, 157, 165, 181, 189, 205, 208, 213, 229, 237, 241, 252, 253, 261, 277, 285, 300, 301, 309, 325, 328, 333, 340, 349, 357, 360, 373, 381, 397, 405, 421, 429, 445, 453, 468, 469, 477

Offset: 1

Allan Wilks

nonn

Cf. A042944, A042945, A042946, A045674, A112652.

Offset 1 and name corrected by Michel Marcus, Sep 13 2019

cp's OEIS Frontend

A042945 Numbers congruent to 2,3,6,11 mod 12 missing from A042944 (conjectured to be finite).

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A042946 Frequency of occurrence of numbers appearing in A042944.

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A294510 Residues modulo 24 of curvatures in the Apollonian circle packing A042944.

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A248938 Decimal expansion of beta = G^2*(2/3)*Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)) (where G is Catalan's constant), a constant related to the problem of integral Apollonian circle packings.

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A189226 Curvatures in the nickel-dime-quarter Apollonian circle packing, ordered first by generation and then by size.

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A045506 Inscribe 2 spheres of curvature 2 inside sphere of curvature -1, continue to inscribe spheres where possible; sequence gives list of curvatures.

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A060790 Inscribe two circles of curvature 2 inside a circle of curvature -1. Sequence gives curvatures of the smallest circles that can be sequentially inscribed in such a diagram.

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A218155 Numbers congruent to 2, 3, 6, 11 mod 12.

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A045673 Curvatures in diagram constructed by inscribing 2 circles of curvature 0 and 1 inside circle of curvature 0, continuing indefinitely to inscribe circles wherever possible.

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A248938 Decimal expansion of beta = G^2(2/3)Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)) (where G is Catalan's constant), a constant related to the problem of integral Apollonian circle packings.