Kellen Myers - cp's OEIS frontend

A274161 Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy.

2, 3, 7, 11, 17, 23, 27, 31, 37, 41, 45, 57, 61, 65, 75, 79, 91, 95, 99, 109, 113, 125, 129, 133, 143, 147, 159, 163, 167, 177, 181, 193, 197, 201, 211, 215, 227, 231, 235, 245, 249, 261, 265, 269, 279, 283, 295, 299, 303, 313, 317, 329, 333, 337, 347, 351

Offset: 1

Jennifer A. Moon & Kellen Myers, Jun 17 2016

The edge-delete game on the graph G is as follows: Two players alternate turns, permanently deleting one edge from G on each turn. The game ends when a vertex is isolated in what remains of G. The player whose deletion created the isolated vertex loses the game.
Here P_{n} refers to the path with n vertices, not n edges.
a(n)-1 gives the zeros in the nim-sequence of octal game .4, see A002187. - Zachary Winkeler, Jul 10 2016

			The number 7 is included because a path on 7 vertices has no winning strategy for player 1 (P1). Consider the edges labeled 1 through 6, left to right along the path. Without loss of generality, P1's first turn is 1, 2, or 3. P1 cannot delete 1 (an immediate loss). If P1 deletes 2, P2 deletes 4 or 5 to force an immediate loss on P1's next turn. If P1 deletes 3, P2 deletes 5 to force the loss.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Pratik Alladi, Neel Bhalla, Tanya Khovanova, Nathan Sheffield, Eddie Song, William Sun, Andrew The, Alan Wang, Naor Wiesel, Kevin Zhang Kevin Zhao, PRIMES STEP Plays Games, arXiv:1707.07201 [math.CO], 2017, Section 8.
R. P. Gallant, G. Gunther, B. L. Hartnell, D. F. Rall, A game of edge removal on graphs, JCMCC, 57 (2006), 75 - 82.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1)

Cf. A002187.

Union[{2, 3, 17, 37}, Flatten[Outer[Plus, {7, 11, 23, 27, 31}, 34 Range[0, 10]]]]

a(n)=if(n<10, [2, 3, 7, 11, 17, 23, 27, 31, 37][n], [7, 11, 23, 27, 31][n%5+1] + (34*(n\5-1))); \\ Andrew Howroyd, Nov 11 2018

The sequence consists of {2,3,17,37} along with all positive integers congruent to 7, 11, 23, 27, and 31 modulo 34.

a(n) = a(n-1) + a(n-5) - a(n-6) for n > 15. - Andrew Howroyd, Nov 11 2018

A258973 The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al., where zeros have no weight.

Original entry on oeis.org

1, 3, 10, 40, 181, 884, 4539, 24142, 131821, 734577, 4160626, 23881695, 138610418, 812104884, 4796598619, 28529555072, 170733683579, 1027293807083, 6211002743144, 37713907549066, 229894166951757, 1406310771154682, 8630254073158599, 53117142215866687, 327800429456036588

Offset: 0

Kellen Myers, Jun 15 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, Combinatorics of λ-terms: a natural approach, arXiv:1609.08106 [cs.LO], 2016.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, A Natural Counting of Lambda Terms, arXiv preprint arXiv:1506.02367 [cs.LO], 2015.
Maciej Bendkowski and Pierre Lescanne, On the enumeration of closures and environments with an application to random generation, Logical Methods in Computer Science (2019) Vol. 15, No. 4, 3:1-3:21.
K. Grygiel and P. Lescanne, A natural counting of lambda terms, Preprint 2015.

Cf. A086616, A105633, A114851, A360102.

a:= proc(n) option remember; `if`(n<4, [1, 3, 10, 40][n+1],
      ((8*n-3)*a(n-1)-(10*n-13)*a(n-2)
     +(4*n-11)*a(n-3)-(n-4)*a(n-4))/(n+1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 30 2015
a := n -> add(hypergeom([(i+1)/2, i/2+1, i-n+1], [1, 2], -4), i=0..n-1):
seq(simplify(a(n)), n=0..25); # Peter Luschny, May 03 2018

a[n_] := a[n] = If[n<4, {1, 3, 10, 40}[[n+1]], ((8*n-3)*a[n-1] - (10*n-13)*a[n-2] + (4*n-11)*a[n-3] - (n-4)*a[n-4])/(n+1)]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 22 2015, after Alois P. Heinz *)

a(n):=sum(sum((binomial(k+i-1,k-1)*binomial(2*k+i-2,k+i-1)*binomial(n-i-1,n-k-i))/k,k,1,n-i),i,0,n); /* Vladimir Kruchinin, May 03 2018 */

lista(nn) = {z = y + O(y^nn); Vec(((1-z)^2 - sqrt((1-z)^4-4*z*(1-z))) / (2*z*(1-z)));} \\ Michel Marcus, Jun 30 2015

G.f. G(z) satisfies z*G(z)^2 - (1-z)*G(z) + 1/(1-z) = 0 (see Bendkowski link Appendix B, p. 23). - Michel Marcus, Jun 30 2015
a(n) ~ 3^(n+1/2) * sqrt(43/(2*((43*(3397 - 261*sqrt(129)))^(1/3) + (43*(3397 + 261*sqrt(129)))^(1/3) - 86)*Pi)) / (3 - (2*6^(2/3)) / (sqrt(129)-9)^(1/3) + (6*(sqrt(129)-9))^(1/3))^n / (2*n^(3/2)). - Vaclav Kotesovec, Jul 01 2015
a(n) = 1 + a(n-1) + Sum_{i=0..n-1} a(i)*a(n-1-i). - Vladimir Kruchinin, May 03 2018
a(n) = Sum_{i=0..n} Sum_{k=1..n-i} binomial(k+i-1,k-1)*binomial(2*k+i-2,k+i-1)*binomial(n-i-1,n-k-i)/k. - Vladimir Kruchinin, May 03 2018
a(n) = Sum_{i=0..n-1} hypergeom([(i+1)/2, i/2+1, i-n+1], [1, 2], -4). - Peter Luschny, May 03 2018
From Peter Bala, Sep 02 2024: (Start)
a(n) = Sum_{k = 0..n} 1/(k + 1) * binomial(2*k, k)*binomial(n+2*k+1, 3*k+1).
Partial sums of A360102. Cf. A086616.
a(n) = (n + 1)*hypergeom([1/2, -n, (n+2)/2, (n+3)/2], [2, 2/3, 4/3], -16/27).
P-recursive: (n + 1)*a(n) = (8*n - 3)*a(n-1) - (10*n - 13)*a(n-2) + (4*n - 11)*a(n-3) - (n - 4)*a(n-4) with a(0) = 1, a(1) = 3, a(2) = 10 and a(3) = 40.
G.f. A(x) = 1/(1 - x)^2 * c(x/(1-x)^3) = (1 - x - sqrt((1 - 7*x + 3*x^2 - x^3)/(1 - x)))/(2*x), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

More terms from Michel Marcus, Jun 30 2015

A254731 Number of ON cells in the even-rule cellular automaton after n steps with the Moore neighborhood (8 neighbors), with minimal nontrivial symmetric initial state (0,0), (0,1), (1,0), and (1,1) ON.

Original entry on oeis.org

4, 8, 24, 20, 32, 68, 48, 72, 116, 88, 104, 140, 188, 160, 284, 272, 268, 320, 372, 352, 496, 488, 524, 608, 556, 628, 692, 820, 764, 808, 864, 976, 1024, 920, 1032, 1228, 1188, 1256, 1408, 1496, 1488, 1564, 1584, 1712, 1752, 1708, 1888, 2148, 2040, 2100, 2308, 2392, 2544, 2480, 2760, 2752, 2764, 3064, 3020, 2976, 3516, 3440, 3560, 3580, 3804, 3816, 3916, 4236, 4492, 4340, 4516, 4512, 4984, 4764, 5004, 4880, 5116, 5716, 5540, 5560, 5564, 5840, 6200, 6368, 6280, 6668, 6880, 6908, 6960, 7600, 7388, 7396, 8028, 7832, 8332, 8152, 8268, 8928, 8708, 9144

Offset: 0

Kellen Myers, Feb 06 2015

nonn

The rule turns a cell to ON at step n if an even, nonzero number of its eight neighbors were ON in the previous. For example, at n=2 the cell (0,0) is ON because the two neighbors (-1,0) and (0,-1) and no others were ON at the previous step.

It appears that whenever n is divisible by 3, there is a visible disjoint 2x2 square leading the automaton in each cardinal direction.

			For n=3, the configuration includes the initial four ON cells plus four other 2 X 2 squares in each cardinal direction.

Index entries for sequences related to cellular automata

Cf. A160239.

m = 100; n = 2 m + 1;
A = Table[0, {p, 1, m}, {q, 1, n}, {z, 1, n}];
A[[1, m, m + 1]] = 1;
A[[1, m, m]] = 1;
A[[1, m + 1, m + 1]] = 1;
A[[1, m + 1, m]] = 1;
For[i = 2, i <= m, i++,
For[x = 2, x <= n - 1, x++,
  For[y = 2, y <= n - 1, y++,
   sum = A[[i - 1, x - 1, y - 1]] +
     A[[i - 1, x, y - 1]] +
     A[[i - 1, x + 1, y - 1]] +
     A[[i - 1, x - 1, y]] +
     A[[i - 1, x + 1, y]] +
     A[[i - 1, x - 1, y + 1]] +
     A[[i - 1, x, y + 1]] +
     A[[i - 1, x + 1, y + 1]];
   A[[i, x, y]] = If[sum > 0, 1 - Mod[sum, 2], 0];
   ]
  ]
];
Table[Plus @@ Plus @@ A[[i, All, All]], {i, 1, m}]
(* Kellen Myers, Feb 07 2015 *)

A250026 The 2-color Rado numbers for x_1^2 + x_2^2 + ... + x_n^2 = z^2.

Original entry on oeis.org

1, 7825, 105, 37, 23, 18, 20, 20, 15, 16, 20, 23, 17, 21, 26, 17, 23, 28, 25, 29, 29, 26, 36, 32, 27, 38, 33, 35, 41, 36

Offset: 1

Kellen Myers, Nov 10 2014

The value of a(2) was only recently discovered (see Heule, Kullmann, & Marek link). - Kellen Myers, May 27 2016

			The integers 1 through 105 cannot be 2-colored without inducing a monochromatic solution to x^2+y^2+w^2=z^2 (and 105 is the least such number), thus a(3)=105.

Paul Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Université de Genève, L'Enseignement Mathématique 28 (1980).

Marijn J. H. Heule, Oliver Kullmann, Victor W. Marek, Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer arXiv:1605.00723 [math.CO], May 2016.
Kellen Myers, A Note on a Question of Erdős & Graham, arXiv:1501.05085 [math.CO], Jan 2015.
Kellen Myers and Joseph Parrish, Some Nonlinear Rado Numbers, Integers, 18B (2018), #A6.

a(18)-a(30) from Kellen Myers, Mar 17 2015

a(1)-a(2) from Kellen Myers, May 27 2016

A162672 Lunar product 19*n.

Original entry on oeis.org

0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150

Offset: 0

Emilie Hogan, Dennis Hou, Kellen Myers and N. J. A. Sloane, Apr 09 2010

Since 19 is the smallest lunar prime, this is a kind of lunar analog of the even numbers.

As the b-file shows, this sequence is not monotonic and contains repetitions.

			19 * 3 = 13, so 13 is a member. 1109 has just two divisors, 9 and 109, so 1109 is not a member.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

Cf. A087062, A078645.

For a two-digit number n, the lunar product 19*n is obtained by putting a 1 in front of n.

Entry revised by N. J. A. Sloane, May 28 2011, to correct errors in some of the comments

A177101 The number of cycles in the Vers de Verres game, where 'worms' are transferred between 'cups' in a deterministic fashion. Because this defines a finite-state automaton, we know that every state eventually enters a cycle (or fixed point, which is essentially a cycle of length 1). The number of 'cups' (frequently called 'n') is a parameter for this automaton, and so we count the cycles (and fixed points) with respect to n.

Original entry on oeis.org

1, 2, 4, 7, 13, 14, 20

Offset: 1

Kellen Myers, May 02 2010

The game is described in the websites listed, and already has other sequences, e.g., A151986. Note that this also gives the number of connected components, if we draw a graph of this process. The sequence gives the number of cycles, for a given number of cups. The sequence is increasing (append a 0 to all configurations in a cycle, and you get the same cycle with one more cup). It is strictly increasing since {n-1,0,0,0...,0} occurs in a cycle at stage n, but never before.

I am not clear on how this is meant to differ from A176450; my calculations reproduce the terms there not the ones in this sequence. - Joseph Myers, Nov 13 2010

			For n=4, there are seven cycles: {0300,3000,0030}, {3300,3003,0330}, {0200,2000}, {3330}, {2200}, {1000}, {0000}. Note that four of these are "inherited" from n=3, as described above.

Eric Angelini - Vers de Verres
E. Angelini, Vers de verres (Glass worms) [Cached copy, with permission]
Kellen Myers - Vers de Verres [Broken link]

Related to A151986, A151987, A176336.

Fixed error in sequence. Added small amount of formatting changes and elaboration. - Kellen Myers, May 03 2010

cp's OEIS Frontend

User: Kellen Myers

A274161 Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy.

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A258973 The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al., where zeros have no weight.

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A254731 Number of ON cells in the even-rule cellular automaton after n steps with the Moore neighborhood (8 neighbors), with minimal nontrivial symmetric initial state (0,0), (0,1), (1,0), and (1,1) ON.

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A250026 The 2-color Rado numbers for x_1^2 + x_2^2 + ... + x_n^2 = z^2.

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A162672 Lunar product 19*n.

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