Adam P. Goucher - cp's OEIS frontend

A308634 Number of triangle-free acyclic digraphs (or DAGs) up to isomorphism with n vertices, maximum indegree 2 and unique maximal element.

1, 1, 2, 5, 18, 81, 475, 3394, 29140, 293198

Offset: 1

Adam P. Goucher, Jun 12 2019

			a(1) = 1, a(2) = 1 and a(3) = 2 coincide with the first three terms of A001190, as all connected triangle-free DAGs with at most three vertices are necessarily trees.
For n = 4, the a(4) = 5 solutions are the following three trees:
(((())))
((), (()))
(((), ()))
together with two 4-cycles.

Lower-bounded by A001190.

A257319 Numbers n such that the n-th generation of Sawtooth 201 has minimum population in Conway's Game of Life.

Original entry on oeis.org

0, 1840, 88320, 4152880, 195187200, 9173800240, 431168613120, 20264924818480, 952451466470400, 44765218924110640, 2103965289433201920, 98886368603360492080, 4647659324357943129600, 218439988244823327093040, 10266679447506696373374720

Offset: 1

Adam P. Goucher, Apr 20 2015

			The pattern begins at its minimum population of 201 at generation 0, and first returns to this population at generation 1840.

LifeWiki, Sawtooth 201
Index entries for linear recurrences with constant coefficients, signature (48,-47).

Cf. A218750.

NestList[(47 # + 1840) &, 0, 100]
LinearRecurrence[{48,-47},{0,1840},20](* Harvey P. Dale, Sep 19 2017 *)

a(n) = 47*a(n-1) + 1840.
a(n) = (47^(n-1) - 1)*40. - Bill Gosper, Apr 25 2015
G.f.: 1840*x^2/((47*x-1)*(x-1)). - Alois P. Heinz, May 03 2015
E.g.f.: 40*(exp(47*x) - 47*exp(x) + 46)/47. - Stefano Spezia, Jul 04 2022
a(n) = 1840 * A218750(n-1). - Alois P. Heinz, Jul 04 2022

A249510 Ménage primes.

Original entry on oeis.org

2, 13, 775596313, 567525075138663383127158192994765939404930668817780601409606090331861

Offset: 1

Adam P. Goucher, Oct 30 2014

nonn

The fifth ménage prime has 30281 digits and is therefore too large to include here.

			The fifth ménage number is 13, which is prime.

Adam P. Goucher, Menage primes, Complex Projective 4-Space.

Intersection of A000040 (primes) with A000179 (ménage numbers).

Image of A249394 under A000179.

Comment correction by Hans Havermann, Jan 19 2019

A249394 Numbers k such that the k-th ménage number is prime.

Original entry on oeis.org

4, 5, 13, 53, 8645

Offset: 1

Adam P. Goucher, Oct 30 2014

This sequence is heuristically conjectured to grow doubly-exponentially.

a(6) > 100000. - Hans Havermann, Jan 19 2019

			The only known ménage primes are M(4) = 2, M(5) = 13, M(13) = 775596313, M(53) = 567525075138663383127158192994765939404930668817780601409606090331861, and M(8645).

Adam P. Goucher, Menage primes, Complex Projective 4-Space.

By definition, the inverse image of A000040 under the function A000179.

The ménage primes themselves are given in A249510.

is(n)=ispseudoprime(round(2*n*exp(-2)*besselk(n, 2))) \\ Charles R Greathouse IV, Nov 10 2014

Example edited by Hans Havermann, Jan 19 2019

A229031 Number of 5-colorings of the strong product of the complete graph K2 and the cycle graph Cn.

Original entry on oeis.org

120, 0, 2400, 3840, 63360, 215040, 1943040, 9031680, 64665600, 346030080, 2243911680, 12792299520, 79437987840, 465890181120, 2838290104320, 16857940623360, 101834835886080, 608260231004160, 3660556491816960, 21919358464819200, 131692072607416320, 789448748118835200, 4739507238312345600, 28425784430470103040

Offset: 2

Adam P. Goucher, Sep 11 2013

The strong product of K2 and Cn can be regarded as the King's graph on a 2*n cylindrical (or equivalently toroidal) chessboard.
The Kneser graph construction of the Petersen graph relates this to the number of closed walks on the Petersen graph.
More generally, the number of c-colorings of the strong product of Km and Cn is equal to (m!)^n * (c choose m) * (number of closed walks of length n on K(c,m)).
If n is prime then a(n) is divisible by n, since the cyclic group of order n acts on the colorings, partitioning them into orbits of size n. More generally, n divides a(n) for any Carmichael number n, due to the closed form.

			For n = 2, the graph is the complete graph K4, which has a(4) = 120 different 5-colorings corresponding to ordered 4-subsets of {1,2,3,4,5}.
For n = 3, the graph is the complete graph K6, which cannot be 5-colored, so a(3) = 0. Equivalently, there are no closed walks of length 3 on the Petersen graph.

Indranil Ghosh, Table of n, a(n) for n = 2..1000
Wolfram MathWorld, Graph Strong Product
Index entries for linear recurrences with constant coefficients, signature (4,20,-48).

Table[2^n(3^n+4(-2)^n+5),{n,2,25}]
LinearRecurrence[{4,20,-48},{120,0,2400},24] (* or *) Drop[CoefficientList[Series[-120*x^2*(4*x - 1) / ((2*x - 1) * (4*x + 1) * (6*x - 1)), {x, 0, 25}], x], 2] (* Indranil Ghosh, Mar 03 2017 *)

a(n) = (2^n) * (3^n + 4*(-2)^n + 5) \\ Indranil Ghosh, Mar 03 2017

def A229031(n) : return (2**n) * (3**n + 4*(-2)**n +5) # Indranil Ghosh, Mar 03 2017

a(n) = 6^n + 4*(-4)^n + 5*2^n.
a(n) = 10 * 2^n * A091000(n).
a(n) = 4*a(n-1)+20*a(n-2)-48*a(n-3). G.f.: -120*x^2*(4*x-1) / ((2*x-1)*(4*x+1)*(6*x-1)). - Colin Barker, Oct 20 2013

A228649 Numbers n such that n-1, n and n+1 are all squarefree.

Original entry on oeis.org

2, 6, 14, 22, 30, 34, 38, 42, 58, 66, 70, 78, 86, 94, 102, 106, 110, 114, 130, 138, 142, 158, 166, 178, 182, 186, 194, 202, 210, 214, 218, 222, 230, 238, 254, 258, 266, 282, 286, 302, 310, 318, 322, 330, 346, 354, 358, 366, 382, 390, 394, 398, 402, 410, 418, 430, 434, 438, 446, 454, 462, 466, 470, 482, 498

Offset: 1

Adam P. Goucher, Aug 29 2013

Equivalently, a positive integer n is comfortably squarefree if and only if n^3 - n is squarefree. The 'if' direction is obvious from the factorization n(n-1)(n+1), and the converse follows from the coprimality of n, n - 1 and n + 1.
The asymptotic density of comfortably squarefree numbers is the product over all primes of 1 - 3/p^2, which is A206256 = 0.125486980905....
See also comments in A007675.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Dan Asimov, Interesting sequence on MathOverflow, Math-fun mailing list, Mar 28 2023.
Ewan Delanoy, Are there infinitely many triples of consecutive square-free integers?, Math Overflow.
A. P. Goucher, Comfortably squarefree numbers, Complex Projective 4-Space.
Fredrick M. Nelson, Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, Mar 24 2023.

with(numtheory):
a := n -> `if`(issqrfree(n-1) and issqrfree(n) and issqrfree(n+1), n, NULL);
seq(a(n), n = 1..500); # Peter Luschny, Jan 18 2014

Select[Range[500], (SquareFreeQ[# - 1] && SquareFreeQ[#] && SquareFreeQ[# + 1]) &] (* Adam P. Goucher *)
Select[Range[2, 500, 2], (MoebiusMu[# - 1] MoebiusMu[#] MoebiusMu[# + 1]) != 0 &] (* Alonso del Arte, Jan 16 2014 *)
Flatten[Position[Partition[Boole[SquareFreeQ/@Range[500]],3,1],{1,1,1}]]+1 (* Harvey P. Dale, Jan 14 2015 *)
SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,500}],{1,1,1}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 02 2018 *)

is(n)=issquarefree(n-1)&&issquarefree(n)&&issquarefree(n+1) \\ Charles R Greathouse IV, Aug 29 2013

a(n) = A007675(n) + 1. - Giovanni Resta, Aug 29 2013

A193362 Numbers of ways in which a unit disc can be dissected into 6n curvilinear triangles, at least one of which does not contain the center.

Original entry on oeis.org

0, 31, 57, 99, 158, 237, 340, 472, 635, 836, 1075, 1361, 1696, 2087, 2538, 3054, 3641, 4306, 5053, 5891, 6822, 7857, 9000, 10260, 11643, 13156, 14807, 16605, 18556, 20671, 22954, 25418, 28069, 30918, 33973, 37243, 40738, 44469, 48444, 52676

Offset: 1

Adam P. Goucher, Dec 20 2012

			For n = 2, the a(2) = 31 dissections of the disc into 6n = 12 curvilinear triangles are:
* 1 solution in which 1 piece does not touch the center;
* 5 solutions in which 2 pieces do not touch the center;
* 10 solutions in which 3 pieces do not touch the center;
* 10 solutions in which 4 pieces do not touch the center;
* 3 solutions in which 5 pieces do not touch the center;
* 2 symmetrical solutions, one of which is exceptional.
The 30 non-exceptional cases are given in the article 'Dissecting the disc'.

H. T. Croft, K. J. Falconer and R. K. Guy, "Unsolved Problems in Geometry", 1991, page 89.

A. P. Goucher, Dissecting the disc, Complex Projective 4-Space.

Table[If[n==1,0,Boole[n==2]+1+2 n+1+(3 n^2+3 n+2)/2+Floor[(2 n^3+6 n^2+7 n+6)/6]+Floor[(n^4+10 n^3+35 n^2+50 n+120)/120]+1],{n,1,100}]

A215878 Lengths of loops in the P2 Penrose tiling.

Original entry on oeis.org

10, 20, 80, 100, 460, 620, 2780, 3700, 16660, 22220, 99980, 133300, 599860, 799820, 3599180, 4798900, 21595060, 28793420, 129570380, 172760500, 777422260, 1036563020, 4664533580, 6219378100, 27987201460, 37316268620, 167923208780, 223897611700, 1007539252660, 1343385670220, 6045235515980

Offset: 1

Adam P. Goucher, Aug 25 2012

A loop of length n is defined to be an ordered set of n tiles (kites or darts), such that the tile T_i shares an edge with each of T_(i+1) and T_(i-1) (subscripts considered modulo n), but does not share a vertex with any other tile in the loop. These loops are the finite paths traced by gliders in a particular cellular automaton on the P2 Penrose tiling.

			The smallest loop a(1)=10 corresponds to the 10 kites which form the perimeter of a regular decagon.

Jacob Aron, First gliders navigate ever-changing Penrose universe, New Scientist.
Adam P. Goucher, Blog post about this
Adam P. Goucher, Gliders in Cellular Automata on Penrose Tilings, Journal of Cellular Automata (2012).
Index entries for linear recurrences with constant coefficients, signature (0,5,0,6).

Table[{1,1}.MatrixPower[{{5,2},{3,0}},Floor[n/2]].{10,10Mod[n,2]},{n,0,49}]
Table[-(5/7)(-6^(1/2(n-1))(9+2Sqrt[6]+(-1)^n(-9+2Sqrt[6]))+4(Cos[n Pi/2] + Sin[n Pi/2])), {n, 1, 20}] (* Benedict W. J. Irwin, Nov 01 2016 *)

Vec(-10*x*(3*x^2+2*x+1)/((x^2+1)*(6*x^2-1)) + O(x^100)) \\ Colin Barker, May 19 2014

Recurrence relation: a(n+4) = 5*a(n+2) + 6*a(n).
G.f.: -10*x*(3*x^2+2*x+1) / ((x^2+1)*(6*x^2-1)). - Colin Barker, May 19 2014
a(n) = 3*a(n-1)+2*a(n-2) if n is odd. a(n) = 2*a(n-1)-3*a(n-2) if n is even. - R. J. Mathar, Jun 18 2014
a(n) = -5 * ( -6^((n - 1)/2) * (9 + 2*sqrt(6) + (-1)^n * (2 * sqrt(6) - 9)) + 4 * (cos(n * Pi/2) + sin(n * Pi/2)))/7. - Benedict W. J. Irwin, Nov 01 2016

cp's OEIS Frontend

User: Adam P. Goucher

A308634 Number of triangle-free acyclic digraphs (or DAGs) up to isomorphism with n vertices, maximum indegree 2 and unique maximal element.

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A257319 Numbers n such that the n-th generation of Sawtooth 201 has minimum population in Conway's Game of Life.

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Mathematica

Formula

A249510 Ménage primes.

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A249394 Numbers k such that the k-th ménage number is prime.

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PARI

Extensions

A229031 Number of 5-colorings of the strong product of the complete graph K2 and the cycle graph Cn.

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Mathematica

PARI

Python

Formula

A228649 Numbers n such that n-1, n and n+1 are all squarefree.

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Maple

Mathematica

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A193362 Numbers of ways in which a unit disc can be dissected into 6n curvilinear triangles, at least one of which does not contain the center.

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Mathematica

A215878 Lengths of loops in the P2 Penrose tiling.

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Mathematica

PARI