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A387346 Big Ramsey degree (with respect to substructures) of an independent set of size n in a universal triangle-free graph.

1, 5, 161, 134397, 7980983689, 45921869097999781, 35268888847472944795910097, 4885777205485902177648027702583670093, 159271391109084147116751767705171032995283089412057, 1546604163029698823334234758731306633891622324147639816544352644405

Offset: 1

Jan Hubicka and Stepan Vodsedalek, Aug 27 2025

Let G be any countable universal triangle-free graph. By universality we mean that every countable triangle-free graph is isomorphic to an induced subgraph of G. For every integer n and every finite coloring of the n-element independent sets of G, there exists an induced subgraph H isomorphic to G such that the number of colors used on the n-element independent sets within H is at most a(n). Conversely, there exists a coloring of the n-element independent sets of G with a(n) colors such that every induced subgraph isomorphic to G uses all a(n) colors.

The term big Ramsey degree was introduced by Kechris, Pestov, and Todorcevic (2005). The first element of the sequence was determined by Komjáth and Rödl (1986). The existence of the infinite sequence was proved by Dobrinen (2020) with shorter proofs given by Zucker (2022) and Hubička (2025). The exact sequence was characterized by Balko, Chodounský, Dobrinen, Hubička, Konečný, Vena, and Zucker (2024), with an easier presentation provided by the same authors in arXiv:2303.10088. The first 10 elements of the sequence were computed by Konečný, Hubička, Vodseďálek, and Zucker (2025).

M. Balko, D. Chodounský, N. Dobrinen, J. Hubička, M. Konečný, L. Vena, A. Zucker, Exact big Ramsey degrees for finitely constrained binary free amalgamation classes, Journal of the European Mathematical Society, 2024, published online first.
N. Dobrinen, The Ramsey theory of the universal homogeneous triangle-free graph, Journal of Mathematical Logic, 20 (2020), 2050012.
J. Hubička, Big Ramsey degrees using parameter spaces, Advances in Mathematics 478 (2025), 110386.
J. Hubička, M. Konečný, Š. Vodseďálek, A. Zucker, Counting big Ramsey degrees of the homogeneous and universal K_4-free graph, arXiv:2505.22620, extended abstract accepted to Eurocomb 2025.
A. S. Kechris, V. G. Pestov, S. Todorcevic, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geometric and Functional Analysis 15(1) (2005), 106-189.
P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs and Combinatorics 2(1) (1986) 55-60.
N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math. 185(1-3) (1998) 137-181.
Š. Vodseďálek, Counting big Ramsey degrees, Bachelor's thesis, Charles University (2025).
A. Zucker, On big Ramsey degrees for binary free amalgamation classes, Advances in Mathematics 408 (2022), 108585

M. Balko, D. Chodounský, N. Dobrinen, J. Hubička, M. Konečný, L. Vena, and A. Zucker, Characterisation of the big Ramsey degrees of the generic partial order, arXiv:2303.10088 [math.CO], 2025.
J. Hubička and A. Zucker, A survey on big Ramsey structures, arXiv:2407.17958 [math.LO], 2025.

The corresponding sequence for countable universal graphs (e.g. the Rado/random graph) is A000182.

A387619 Number of ways to tile a 2 X n strip with squares, dominoes, and (straight and bent) trominoes.

Original entry on oeis.org

1, 2, 11, 51, 235, 1092, 5064, 23489, 108954, 505377, 2344171, 10873339, 50435526, 233943074, 1085135139, 5033353844, 23347000765, 108294084141, 502317568673, 2329978980834, 10807509809918, 50130181109907, 232526743191648, 1078565548781339, 5002881075314417

Offset: 0

Greg Dresden and Daeheon Shin, Sep 03 2025

Compare to A030186 which counts tilings with just squares and dominoes, and to A052980 which counts tilings with just dominos and bent trominos.

Index entries for linear recurrences with constant coefficients, signature (3,6,7,2,0,-1).

Cf. A030186, A052980.

LinearRecurrence[{3, 6, 7, 2, 0, -1}, {1, 2, 11, 51, 235, 1092}, 40]

a(n) = 3*a(n-1) + 6*a(n-2) + 7*a(n-3) + 2*a(n-4) - a(n-6).

G.f.: (1 - x - x^2 - x^3)/(1 - 3*x - 6*x^2 - 7*x^3 - 2*x^4 + x^6).

A386305 Numbers of people such that the first person is freed in the variant of the Josephus problem in which one person is skipped, then one is eliminated, then two people are skipped and one eliminated, then three people are skipped and so on.

Original entry on oeis.org

1, 2, 3, 18, 22, 171, 195, 234, 1262, 2136, 6040, 42545, 353067, 1332099, 1447753, 2789475, 3635021, 7857445, 9224024, 17128159, 27666710, 29279638

Offset: 1

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Aug 20 2025

This sequence can also be described in terms of "AP dealing", in which one deals a deck of N cards into a new deck by moving one card to the bottom, dealing out the next card on top of the new deck, moving two cards to the bottom, etc. This sequence consists of all the deck sizes such that the top card remains the same after AP dealing.

Numbers k such that A291317(k) = 1.

			Suppose there are 5 people in a circle. We start with skipping one person and eliminating the next (person number 2). The leftover people are 3,4,5,1 in order. Then we skip two people and eliminate person number 5. The leftover people are 1,3,4 in order. Then we skip three people and person number 1 is eliminated. The leftover people are 3,4 in order. Then we skip four people and eliminate person number 3. Person 4 is freed. As person 1 is not freed, 5 is NOT in this sequence.

Cf. A081614, A291317, A385328, A386312, A386643.

def F(n):
    c, i, J = 1, 0, list(range(1, n+1))
    while len(J) > 1:
        i = (i + c) % len(J)
        q = J.pop(i)
        c = c + 1
    return J[0]
print([n for n in range(1, 100000) if F(n) == 1])

a(20)-a(22) from Jinyuan Wang, Aug 31 2025

A386312 Numbers of people such that the last person is freed in the variant of the Josephus problem in which one person is skipped, then one is eliminated, then two people are skipped and one eliminated, then three people are skipped and so on.

Original entry on oeis.org

1, 7, 10, 12, 21, 25, 28, 235, 822, 24886, 99607, 101497, 107716, 5756103, 55480598

Offset: 1

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Aug 20 2025

This sequence can also be described in terms of "AP dealing", in which one deals a deck of N cards into a new deck by moving one card to the bottom, dealing out the next card on top of the new deck, moving two cards to the bottom, etc. This sequence consists of all the deck sizes such that the bottom card of the deck moves to the top after AP dealing.

Numbers k such that A291317(k) = k.

			Suppose there are people 1,2,3,4,5,6,7 in a circle. We first skip one person and eliminate the next, leaving people in order 3,4,5,6,7,1. Now, we skip two people and eliminate the next, leaving 6,7,1,3,4. Now, we skip three and eliminate the next, leaving 4,6,7,1. Now, we skip four and eliminate the next, leaving 6,7,1. Now, we skip five and eliminate the next, leaving 6,7. Finally, we skip six and eliminate the next, leaving just 7. As the last person in the circle was freed, 7 belongs to this sequence.

Cf. A182459, A291317, A385513, A386305.

def F(n):
    c, i, J = 1, 0, list(range(1, n+1))
    while len(J) > 1:
        i = (i + c) % len(J)
        q = J.pop(i)
        c = c + 1
    return J[0]
print([n for n in range(1, 100000) if F(n) == n])

a(15) from Jinyuan Wang, Aug 31 2025

A386528 Primes which remain primes after the mapping {1 -> 3, 3 -> 5, 5 -> 7, 7 -> 9, 9 -> 1} of its decimal digits.

Original entry on oeis.org

2, 3, 5, 19, 31, 37, 41, 59, 97, 131, 137, 151, 157, 181, 191, 199, 211, 227, 239, 271, 281, 307, 349, 359, 367, 409, 419, 457, 461, 479, 509, 541, 569, 619, 631, 641, 691, 727, 797, 821, 827, 829, 881, 907, 919, 947, 971, 977, 991, 1009, 1021, 1049, 1069, 1087, 1097, 1109, 1151

Offset: 1

Tristan J. Jones and Robert G. Wilson v, Aug 21 2025

Of the 10! possible nontrivial decimal digital mappings, this one was chosen for its inclusion of all the odd numbers and none of the even numbers.

			19 is a term since the mapping produces 31, which is prime;
31 is a term since the mapping produces 53, which is prime.

Cf. A000040, A198047.

fQ[n_] := PrimeQ[ FromDigits[ IntegerDigits[ n] /. {1 -> 3, 3 -> 5, 5 -> 7, 7 -> 9, 9 -> 1}]]; Select[ Prime@ Range@ 200, fQ]

from sympy import isprime
def ok(n): return isprime(n) and isprime(int(str(n).translate(str.maketrans("13579","35791"))))
print([k for k in range(1200) if ok(k)]) # Michael S. Branicky, Aug 24 2025

A386489 Expansion of (1-x)/((1+x+2x^2)(1-4*x+x^2)).

Original entry on oeis.org

1, 2, 7, 30, 109, 402, 1511, 5638, 21021, 78474, 292887, 1093006, 4079181, 15223810, 56815879, 212039702, 791343293, 2953333114, 11021988791, 41134623134, 153516503405, 572931388658, 2138209053735, 7979904827430, 29781410249821, 111145736175722

Offset: 0

Greg Dresden and Madison Lingchen Zhou, Aug 20 2025

a(n) is the number of ways to tile a 2 X n board with squares, dominoes, and L-shaped quadrominoes. Here is one of the a(4)=109 possible tilings of a 2 X 4 board:
    _____
   | | |||
   ||____|
Compare to A030186 which counts the tilings with just squares and dominos.

Index entries for linear recurrences with constant coefficients, signature (3,1,7,-2).

Cf. A030186, A033505.

LinearRecurrence[{3, 1, 7, -2}, {1, 2, 7, 30}, 30]

a(n) = 3*a(n-1) + a(n-2) + 7*a(n-3) - 2*a(n-4).
a(n) = A030186(n) + 2*sum_{i=0..n-2}(A033505(n-i-3)*a(i) + A030186(n-i-3)*(a(i)+2*sum_{j=0..i} a(j)).
a(n) ~ (2 + sqrt(3))^(n+2) / (18 + 4*sqrt(3)). - Vaclav Kotesovec, Aug 21 2025
23*a(n) = -4*A001353(n)+13*A001353(n+1) +10*A001607(n+1)+8*A001607(n) . - R. J. Mathar, Aug 26 2025

A386936 Numbers that can be represented using their digits in the order of appearance, the operations +, -, *, /, ^, and any parentheses.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 343, 736, 1285, 2187, 2592, 2737, 3125, 3685, 3972, 4096, 6455, 11664, 14641, 15552, 15585, 15617, 15618, 15622, 15624, 15626, 15632, 15645, 15655, 15656, 15662, 15667, 15698, 16377, 16384, 17536, 19683, 23328, 24576, 27639

Offset: 1

Anuraag Pasula and Walter Robinson, Aug 09 2025

Each digit is its own operand (no concatenation of digits).
Real and imaginary intermediate values are allowed as long as the final value of the expression is an integer.
Unary minus is not allowed, otherwise we would have 127 = -1 + 2^7. - Sean A. Irvine, Aug 31 2025

			343 = (3+4)^3.
2737 = (2*7)^3-7.
46688 = (4 + 6^6/8)*8.

Walter Robinson, Expressions for Values up to 50000

Cf. A036057, A046469, A080035, A156954, A362769.

A387151 a(n) = n*n! / Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

0, 1, 2, 3, 8, 10, 12, 14, 64, 216, 240, 264, 576, 624, 672, 720, 6144, 6528, 20736, 21888, 46080, 48384, 50688, 52992, 221184, 1152000, 1198080, 11197440, 23224320, 24053760, 24883200, 25712640, 424673280, 437944320, 451215360, 464486400, 2866544640

Offset: 0

David A. Corneth and Peter Luschny, Aug 18 2025

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..3167

Cf. A387140.

A387151 := n -> n*n! / mul(NumberTheory:-Radical(k), k = 1..n): seq(A387151(n), n = 0..36);

k = 1; {0}~Join~Reap[Do[k *= Times @@ FactorInteger[n][[;; , 1]]; Sow[n*n!/k], {n, 36}] ][[-1, 1]] (* Michael De Vlieger, Aug 18 2025 *)

a(n) = n! / A387140(n) for n >= 1.

A386222 Number of 3-dimensional tilings of a 2 X 2 X (n+1) box with the two upper right cells removed, using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.

Original entry on oeis.org

1, 5, 34, 201, 1241, 7538, 46045, 280693, 1712338, 10443297, 63697825, 388506066, 2369604597, 14452808029, 88151396594, 537657790873, 3279312211305, 20001361622066, 121993408939853, 744068928339589, 4538266259447698, 27680043927136849, 168827650973959281

Offset: 0

Greg Dresden and Xiaoya Gao, Aug 13 2025

Here is the box for n=3:
    __________
   /   /   /   /|
  /_/___/_/ |__
 /   /   /   /| /   /|
/_/___/_/ |/_/ |
|   |   |   | /   /| /
|_|___|_|/_/ |/
|   |   |   |   |  /
|_|___|_|___| /.

			Here is one of the a(1)=5 ways to tile the shape for n=1, in this case with one flat plate on the bottom and one domino on top.
    ____
   /   /|
  /   / |____
 /   /  /   /|
/___/  /   / |
|   | /   /  /
|___|/___/  /
|       |  /
|_______| /.

Index entries for linear recurrences with constant coefficients, signature (5,9,-14).

Cf. A359884

LinearRecurrence[{5, 9, -14}, {1, 5, 34}, 30]

G.f.: 1/(1 - 5*x - 9*x^2 + 14*x^3).
a(n) = 5*a(n-1) + 9*a(n-2) - 14*a(n-3) for n >= 3.
a(n) = A359884(n) + 2*a(n-1).

A387093 Composite numbers that contain only prime digits and whose prime factors contain only prime digits.

Original entry on oeis.org

25, 27, 32, 35, 72, 75, 222, 225, 252, 322, 333, 375, 525, 552, 555, 575, 735, 777, 2352, 2553, 2555, 2775, 3357, 3375, 3552, 3577, 5222, 5352, 5575, 7252, 7322, 23253, 23373, 23532, 23535, 23552, 25275, 25725, 25737, 27232, 27252, 27375, 32352, 32375

Offset: 1

Scott R. Shannon and Ursula Ponting, Aug 16 2025

			25725 is a term as 25725 = 3 * 5^2 * 7^3, and both the number and its prime factors only contain prime digits.

Scott R. Shannon, Table of n, a(n) for n = 1..2298 (all terms < 10^12).

Cf. A061371, A027746, A000040, A019546.

Select[Select[Range[33000], CompositeQ], And[AllTrue[Union@ IntegerDigits[#], PrimeQ], AllTrue[Union@ Flatten@ Map[IntegerDigits, FactorInteger[#][[All, 1]] ], PrimeQ]] &] (* Michael De Vlieger, Aug 16 2025 *)

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A387346 Big Ramsey degree (with respect to substructures) of an independent set of size n in a universal triangle-free graph.

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A387619 Number of ways to tile a 2 X n strip with squares, dominoes, and (straight and bent) trominoes.

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A386305 Numbers of people such that the first person is freed in the variant of the Josephus problem in which one person is skipped, then one is eliminated, then two people are skipped and one eliminated, then three people are skipped and so on.

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A386312 Numbers of people such that the last person is freed in the variant of the Josephus problem in which one person is skipped, then one is eliminated, then two people are skipped and one eliminated, then three people are skipped and so on.

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A386528 Primes which remain primes after the mapping {1 -> 3, 3 -> 5, 5 -> 7, 7 -> 9, 9 -> 1} of its decimal digits.

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A386489 Expansion of (1-x)/((1+x+2*x^2)*(1-4*x+x^2)).

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A386936 Numbers that can be represented using their digits in the order of appearance, the operations +, -, *, /, ^, and any parentheses.

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A387151 a(n) = n*n! / Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

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Maple

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A386222 Number of 3-dimensional tilings of a 2 X 2 X (n+1) box with the two upper right cells removed, using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.

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A386489 Expansion of (1-x)/((1+x+2x^2)(1-4*x+x^2)).