A045506 -id:A045506 - cp's OEIS frontend

A042944 Curvatures in diagram constructed by inscribing 2 circles of curvature 2 inside circle of curvature -1, continuing indefinitely to inscribe circles wherever possible.

-1, 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, 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, 162, 167, 170, 171, 174, 179, 182, 183

Offset: 1

Brian Galebach, Allan Wilks

The sequence seems to follow a pattern where differences between consecutive terms are 3,1,3,5,3,1,3,5,..., which would give A218155. However, some curvatures (starting with 78, listed in A042945) are in that sequence, but missing from the circle diagram.

Clifford A. Pickover, The Mathematics of OZ, Mental Gymnastics From Beyond The Edge, Cambridge University Press, Chapter 104 'Circle Mathematics,' figure courtesy of Allan Wilks, Cambridge, UK, 2002, pages 219-220.

James Spahlinger, Table of n, a(n) for n = 1..10000
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.
I. Peterson, Circle Game, Science News, 4/21/01.
Eric Weisstein's World of Mathematics, Bowl of Integers.

Cf. A042945, A042946, A045506, A218155.

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

A294732 Maximal diameter of the connected cubic graphs on 2*n vertices.

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

Offset: 2

Hugo Pfoertner, Dec 13 2017

			a(5)=5 because there exists a unique graph (up to permutations) on 2*5=10 nodes, given by the following adjacency matrix
.
      1 2 3 4 5 6 7 8 9 10
   1  . 1 1 1 . . . . . .
   2  1 . 1 1 . . . . . .
   3  1 1 . . 1 . . . . .
   4  1 1 . . 1 . . . . .
   5  . . 1 1 . 1 . . . .
   6  . . . . 1 . 1 1 . .
   7  . . . . . 1 . . 1 1
   8  . . . . . 1 . . 1 1
   9  . . . . . . 1 1 . 1
  10  . . . . . . 1 1 1 .
.
that requires a shortest possible walk using 5 edges to get from node 1 to node 9.
From _Andrew Howroyd_, Dec 15 2017: (Start)
The following constructions are optimal (see theorem 5 of Caccetta et al.).
Pattern for odd n >= 10. Each additional 4 nodes increases diameter by 3.
      o         o         o         o
    / | \     / | \     / | \     / | \
   o--o  o---o  |  o---o  |  o---o  o--o
    \ | /     \ | /     \ | /     \ | /
      o         o         o         o
Pattern for even n >= 12. Each additional 4 nodes increases diameter by 3.
      o--o         o         o         o
    / |  | \     / | \     / | \     / | \
   o--o  |  o---o  |  o---o  |  o---o  o--o
    \ |  | /     \ | /     \ | /     \ | /
      o--o         o         o         o
(End)

Colin Barker, Table of n, a(n) for n = 2..1000
L. Caccetta and W. F. Smyth, Graphs of maximum diameter, Discrete Mathematics, Volume 102, Issue 2, 20 May 1992, Pages 121-141.
Gordon Royle, Cubic Graphs, October 1996. [Broken link]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A002851, A204329, A296525, A114119, A045506, A007494.

Apart from initial term, duplicate of A267528.

Prepend[LinearRecurrence[{1, 1, -1}, {2, 3, 5}, 100], 1] (* Jean-François Alcover, Dec 27 2017 *)

Vec(x^2*(1 + x + x^3) / ((1 - x)^2*(1 + x)) + O(x^100)) \\ Colin Barker, Dec 16 2017

From Andrew Howroyd, Dec 15 2017: (Start)
a(n) = (6*n - 11 - (-1)^n)/4 for n > 2.
a(n) = a(n-2) + 3 for n > 4. (End)
From Colin Barker, Dec 16 2017: (Start)
G.f.: x^2*(1 + x + x^3) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 5. (End)
a(n) = A267528(n-1). - Hugo Pfoertner, Oct 10 2018
E.g.f.: (6 + 2*x + x^2 + 3*(x - 2)*cosh(x) + (3*x - 5)*sinh(x))/2. - Stefano Spezia, Feb 20 2023

Terms a(12) and beyond from Andrew Howroyd, Dec 15 2017

A104566 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product H*R of the infinite lower triangular matrices H = [1; 1,2; 1,2,1; 1 2,1,2; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

Original entry on oeis.org

1, 3, 2, 4, 3, 1, 6, 5, 3, 2, 7, 6, 4, 3, 1, 9, 8, 6, 5, 3, 2, 10, 9, 7, 6, 4, 3, 1, 12, 11, 9, 8, 6, 5, 3, 2, 13, 12, 10, 9, 7, 6, 4, 3, 1, 15, 14, 12, 11, 9, 8, 6, 5, 3, 2, 16, 15, 13, 12, 10, 9, 7, 6, 4, 3, 1, 18, 17, 15, 14, 12, 11, 9, 8, 6, 5, 3, 2, 19, 18, 16, 15, 13, 12, 10, 9, 7, 6, 4, 3, 1

Offset: 1

Gary W. Adamson, Mar 15 2005

			The first few rows are
  1;
  3, 2;
  4, 3, 1;
  6, 5, 3, 2;
  ...

Row sums yield A001082.
Columns 1, 3, 5, ... (starting at the diagonal entry) yield A032766.
Columns 2, 4, 6, ... (starting at the diagonal entry) yield A045506.
Row sums = 1, 5, 8, 16, 21, ... (generalized octagonal numbers, A001082). A006578(2n-1) = A001082(2n).

T:=proc(i,j) if j>i then 0 elif i mod 2 = 1 and j mod 2 = 1 then 3*(i-j)/2+1 elif i mod 2 = 0 and j mod 2 = 0 then 3*(i-j)/2+2 elif i+j mod 2 = 1 then 3*(i-j+1)/2 else fi end: for i from 1 to 14 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 24 2005

For 1 <= j <= i: T(i,j) = 3(i-j+1)/2 if i and j are of opposite parity; T(i,j) = 3(i-j)/2 + 1 if both i and j are odd; T(i,j) = 3(i-j)/2 + 2 if both i and j are even. - Emeric Deutsch, Mar 24 2005

More terms from Emeric Deutsch, Mar 24 2005

A114119 Row sums of triangle A114118.

Original entry on oeis.org

1, 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, 105

Offset: 0

Paul Barry, Nov 13 2005

Taken modulo 3 yields 1,0,2,0,2,0,2,0,2,...; a(n) is congruent to 0 or 2 (mod 3) for n > 0.

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

Cf. A007494, A049636, A045506.

Join[{1,3},LinearRecurrence[{1,1,-1},{5,6,8},100]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)

a(n) = 3*floor((n + 1)/2) + 2*((n+1) mod 2) - 0^n.
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(floor((n + k + j)/3), k)*binomial(k, floor((n + k + j)/3)).
G.f.: 1 - x*(-3 - 2*x + 2*x^2)/((1 + x)*(x - 1)^2). - R. J. Mathar, Oct 25 2011
E.g.f.: ((4 + 3*x)*cosh(x) + 3*(1 + x)*sinh(x) - 2)/2. - Stefano Spezia, Feb 20 2023

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A042944 Curvatures in diagram constructed by inscribing 2 circles of curvature 2 inside circle of curvature -1, continuing indefinitely to inscribe circles wherever possible.

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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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A294732 Maximal diameter of the connected cubic graphs on 2*n vertices.

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A104566 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product H*R of the infinite lower triangular matrices H = [1; 1,2; 1,2,1; 1 2,1,2; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

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A114119 Row sums of triangle A114118.

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