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A383447 Number of "peerless" trees on n nodes.

1, 0, 1, 1, 2, 3, 6, 9, 19, 33, 67, 130, 270, 547, 1165, 2456, 5314, 11521, 25357, 56022, 125067, 280471, 633490, 1437340, 3278912, 7510503, 17277697, 39890262, 92427559, 214835923, 500879602, 1171013350, 2744946654, 6450077870

Offset: 1

N. J. A. Sloane, May 01 2025, based on postings to the SeqFan Mailing List in April and May 2025 by Victor S. Miller, Allan C. Wechsler, Brendan McKay, and others

A "peerless" tree is an unlabeled, unrooted tree (as in A000055) with the property that if two nodes are joined by an edge then these nodes have different degrees.
Victor S. Miller reports that this sequence was first proposed on Project Euler.
Comment from Brendan McKay, May 01 2025 (Start)
The enumeration could be extended by the following argument.
If the tree has a unique centroid (not center!) then removing the centroid gives rooted subtrees of size less than n/2. If there are two centroids, they are adjacent and removing that edge gives two rooted subtrees with exactly n/2 vertices.
Start by making all rooted trees up to n/2 vertices which have no adjacent vertices of the same degree, not counting adjacencies of the root. Then classify them according to which degrees the root can be increased to without violating this condition for edges adjacent to the root.
With this information the counts for n vertices can be reconstructed. In this way getting up past 60 vertices should be possible. (End)
This sequence forms the left-most column of A383448.

Victor Miller and others, Peerless trees, Seqfan, 2025.
Project Euler, Problem 936: Peerless Trees.

Cf. A000055, A383448.

a(1)-a(8) were computed by Allan C. Wechsler, Apr 30 2025, and a(9)-a(34) by Brendan McKay, May 02 2025.

A377912 Numbers such that every even digit except the last is immediately followed by a strictly larger digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117

Offset: 1

N. J. A. Sloane, Nov 27 2024, with thanks to Paolo Xausa for correcting the initial definition

In other words, numbers that do not contain a pair of successive digits i, j where i is even and j <= i.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025.

The complement of A347298. Cf. A342042, A377914, A377917, A377918.

A377912Q[k_] := FreeQ[Partition[IntegerDigits[k], 2, 1], {i_?EvenQ, j_} /; j <= i];
Select[Range[0, 200], A377912Q] (* Paolo Xausa, Mar 17 2025 *)

def ok(n):
    s = str(n)
    return not any(s[i] in "2468" and s[i+1] <= s[i] for i in range(len(s)-1))
print([k for k in range(0, 118) if ok(k)]) # Michael S. Branicky, Nov 28 2024

Added 0 to match A342042, and replaced negative definition by a positive one. - N. J. A. Sloane, Nov 29 2024

A376774 Indices of records in A376772.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 25, 28, 30, 37, 40, 41, 50, 55, 57, 66, 67, 76, 85, 93, 94, 139, 148, 157, 165, 174, 175, 179, 188, 197, 269, 278, 279, 288, 297, 369, 378

Offset: 1

N. J. A. Sloane, Nov 05 2024 (with thanks to Michael S. Branicky)

Cf. A302656, A376772, A376773.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

a(37)-a(46) from Dominic McCarty, Nov 08 2024

A376773 Records in A376772.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 21, 25, 44, 48, 52, 53, 58, 75, 77, 84, 96, 135, 146, 300, 317, 401, 452, 478, 1608, 1677, 1679, 1681, 1683, 1703, 1753, 1773, 13649, 13704, 124912, 124925, 125336, 128212, 128221, 128347, 128376, 128529

Offset: 1

N. J. A. Sloane, Nov 05 2024 (with thanks to Michael S. Branicky)

Numbers that are the slowest to appear in A302656.

Cf. A302656, A376772, A376774, A376775.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

a(37)-a(46) from Dominic McCarty, Nov 08 2024

A361337 Numbers that reach 0 after a suitable series of split-and-multiply operations (see Comments for precise definition).

Original entry on oeis.org

0, 10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, 59, 60, 65, 69, 70, 78, 80, 85, 87, 90, 95, 96, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 115, 120, 125, 128, 129, 130, 134, 135, 136, 138, 140, 144, 145, 150, 152, 153, 154, 155, 156, 157, 158, 159

Offset: 1

N. J. A. Sloane, Apr 01 2023, based on a posting to the Sequence Fans mailing list by Eric Angelini, Mar 20 2023

We always split the integer N into two integers, then multiply them (and iterate). For example, 2023 can be split into 20 and 23 (producing 20*23 = 460), or split into 202 and 3 (producing 202*3 = 606). The split 2 and 023 is forbidden, as 023 is not an integer (but 460 can be split into 46 and 0 as 0 is an integer).
The sequence lists numbers which reach 0 after a suitable sequence of splits and multiplications.
If we multiply ALL the digits at each step, we get A034048 (115 is the first term where they differ).
The complement (A361978) appears to be finite, containing only 219 members, the largest being 3111. - Michael S. Branicky, Apr 02 2023
More precisely, {811, 911, 913, 921, 1111, 1112, 1113, 1121, 1122, 1131, 1211, 1231, 1261, 1311, 1321, 1612, 2111, 2121, 2211, 3111} are the only numbers not in the sequence, between 792 and at least 10^7. - M. F. Hasler, Apr 05 2023

			We see that 115 reaches 0 when split into 11*5: 11*5 = 55 -> 5*5 = 25 -> 2*5 = 10 -> 1*0 = 0.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A031346, A034048, A361978.

Supersequence of A011540.

select( {is_A361337(n)=!vecmin(digits(n))|| for(p=1,logint(n,10), is_A361337(vecprod(divrem(n,10^p)))&& return(1))}, [1..160]) \\ M. F. Hasler, Apr 05 2023

def ok(n):
    if n < 10: return n == 0
    s = str(n)
    if "0" in s: return True
    return any(ok(int(s[:i])*int(s[i:])) for i in range(1, len(s)))
print([k for k in range(116) if ok(k)]) # Michael S. Branicky, Apr 02 2023

ok = lambda n: '0' in (s:=str(n)) or any(ok(int(s[:i])*int(s[i:])) for i in range(1,len(s))) # M. F. Hasler, Apr 05 2023

a(2894 + k) = 3112 + k for all k >= 0 (conjectured). - M. F. Hasler, Apr 05 2023

a(38) and beyond from Michael S. Branicky, Apr 02 2023

A361336 Smallest decimal number containing n palindromic substrings (Version 2). See Comments for precise definition.

Original entry on oeis.org

0, 10, 11, 100, 1002, 111, 1000, 10002, 10001, 1111, 10000, 100002, 100001, 1000012, 11111, 100000, 1000002, 1000001, 10000012, 10000010, 111111, 1000000, 10000002, 10000001, 100000012, 100000010, 110111111, 1111111, 10000000, 100000002, 100000001, 1000000012, 1000000010

Offset: 1

N. J. A. Sloane, Apr 01 2023, based on postings to the Sequence Fans Mailing list by Eric Angelini, Mar 28 2023 (definition), and Giovanni Resta, Mar 28 2023 (terms)

Suppose m has decimal expansion d_1 d_2 ... d_k. A palindromic substring here is any substring d_i, d_{i+1}, ..., d_j with 1 <= i <= j <= n which is palindromic. In this version d_i can be 0 even if j>i. For example, if m = 10^3 + 1 = 1001 there are six substrings: 1, 0, 0, 1, 00, and 1001. See A361335 for Version 1.

Giovanni Resta, Table of n, a(n) for n = 1..99

Cf. A361335.

A361335 Smallest decimal number containing n palindromic substrings (Version 1). See Comments for precise definition.

Original entry on oeis.org

0, 10, 11, 101, 1001, 111, 1110, 10101, 10111, 1111, 11110, 102222, 101111, 1001111, 11111, 111110, 1022222, 1011111, 10011111, 11101111, 111111, 1111110, 10222222, 10111111, 100111111, 111111212, 110111111, 1111111, 11111110, 102222222, 101111111, 1001111111, 1111111212, 1101111111, 10101111111, 11111111, 111111110

Offset: 1

N. J. A. Sloane, Apr 01 2023, based on postings to the Sequence Fans Mailing list by Eric Angelini, Mar 28 2023 (definition), and Giovanni Resta, Mar 28 2023 (terms)

Suppose m has decimal expansion d_1 d_2 ... d_k. A palindromic substring here is any substring d_i, d_{i+1}, ..., d_j with 1 <= i <= j <= n which is palindromic, except that if d_i = 0 then i = j. For example, if m = 10^3 + 1 = 1001 there are five substrings: 1, 0, 0, 1, 1001 (but not 00). See A361336 for Version 2.

Giovanni Resta, Table of n, a(n) for n = 1..99

Cf. A361336.

A361625 Number of free polyominoes with checkerboard-pattern-colored vertices with n cells.

Original entry on oeis.org

1, 1, 3, 7, 20, 60, 204, 702, 2526, 9180, 33989, 126713, 476597, 1802109, 6850969, 26151529, 100207548, 385217382, 1485216987, 5741240989, 22246000726, 86383317470, 336093551268, 1309997856337, 5114452295933, 19998171631076, 78306014924606, 307022177714062

Offset: 1

Andrey Zabolotskiy, Mar 19 2023; thanks to John Mason for his help

nonn

Also, number of polysticks of size n (see A019988), with the requirement that any two sticks are connected by a sequence of adjacent, alternately horizontal and vertical sticks. - Pontus von Brömssen, Sep 01 2023

			There are 2 ways to color the 4 corners of a monomino with black and white colors alternatingly, but they are related by a rotation or a reflection, so a(1) = 1. a(2) is also 1 because the two ways to color the 6 vertices of a domino with black and white colors in the checkerboard pattern are related to each other by a reflection or a rotation. The same is true for the stick tromino, but the two ways to color the 8 vertices of the L-tromino are inequivalent, so a(3) = 3.
For n = 3, the a(3) = 3 allowed polysticks are:
  _     _
  _|  _|   _|_

Andrey Zabolotskiy, Table of n, a(n) for n = 1..50
Index entries for sequences related to polyominoes.

A122675 is the 3-dimensional analog based on polycubes.
Cf. A001933, A019988, A359689.
Cf. A000105, A351190, A351142, A351127, A349328, A346799, A234008.
5th row of A366766.

a(n) = 2 * A000105(n) - (A351190(n) + A351142(n) + A351127(n) + A349328(n) + A346799(n/2) + A234008(n/2)), where the last two terms are only included if 2|n. I.e., every free polyomino is counted twice here unless it is symmetric with respect to a Pi/2 rotation centered at a cell, or a Pi rotation centered at an edge, or a reflection with respect to an axis parallel to the grid and passing through cells.

A361502 Index of n-th prime in A359804.

Original entry on oeis.org

2, 3, 4, 8, 13, 42, 347, 3466, 49012, 528231, 717126, 63056215, 1375559400, 7038527851

Offset: 1

N. J. A. Sloane, Mar 18 2023, based on a comment made by Michael De Vlieger in A359804 in which he gave the values of a(1) to a(12)

Theorem: Every prime appears in A359804. For proof see A359804.

It appears that the primes in A359804 appear in order.

Rémy Sigrist, C++ program

Cf. A359804, A361504.

nn = 2^20; c[] = False; q[] = 1;
 i = 1; j = 2; c[1] = c[2] = True; u = 3;
 {2}~Join~Reap[Monitor[Do[
      (k = q[#]; While[c[k #], k++]; k *= #;
         While[c[# q[#]], q[#]++]) &[(p = 2;
        While[Divisible[i j, p], p = NextPrime[p]]; p)];
      If[PrimeQ[k], Sow[n]; Print[n]];
      Set[{c[k], i, j}, {True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}], n]][[-1, -1]] (* Michael De Vlieger, Mar 19 2023 *)

a(13)-a(14) from Rémy Sigrist, Mar 19 2023

A342297 A Van Eck-like sequence based on the Fibonacci numbers. See Comments for definition.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 5, 2, 6, 5, 7, 8, 2, 9, 5, 7, 11, 10, 12, 13, 2, 14, 5, 7, 16, 10, 17, 13, 15, 19, 18, 20, 21, 2, 22, 5, 7, 24, 10, 25, 13, 15, 27, 18, 20, 29, 23, 30, 26, 28, 32, 31, 33, 34, 2, 35, 5, 7, 37, 10, 38, 13, 15, 40, 18, 20, 42, 23, 43, 26, 28, 45, 31, 46, 34, 36, 48

Offset: 1

N. J. A. Sloane, Mar 13 2021, following a suggestion from a correspondent who wishes to remain anonymous. The definition given here is my interpretation of his construction

We construct finite sequences S_0, S_1, S_2, ...; we let A_i  (i >= 0) be the concatenation of S_0, ..., S_i; and the sequence itself is lim_{i -> oo} A_i. All of S_i, A_i, and A have offset 1. F_i (i >= 0) is the i-th Fibonacci number A000045(i).
S_0 = [1] is special.
For i >= 1, S_i has length F_i, and is defined by:
  S_i(j) = max r such that A_{i-1}(r) = j, for 1 <= j <= F_i.
That is, S_i(j) is the index of the most recent occurrence of j in A_{i-1}.
The sequence appears to be as follows. Following the initial 1, it has as a subsequence the Lower Wythoff sequence (A000201: 1, 3, 4, 6, 8, 9, 11, 12, 14, ...) whose terms appear at indices given by the Upper Wythoff sequence (A001950: 2, 5, 7, 10, 13, 15, 18, 20, ...). Interspersed with this is an infinite set of increasingly long initial segments of the Upper Wythoff sequence, each one ending when it has exceeded the most recently appearing term of the Lower Wythoff subsequence. This all shows up clearly on the scatterplot. - Peter Munn, Mar 14 2021

			Here are the initial stages of the construction:
S_0 = [1].
A_0 = S_0 = [1].
S_1: When did 1 last appear in A_0? Answer: r=1, S_1 = [1], A_1 = [1,1].
S_2: When did 1 last appear in A_1? Answer: r=2, S_2 = [2], A_2 = [1,1,2].
S_3: When did 1 last appear in A_2? Answer: r=2.
When did 2 last appear in A_2? Answer: r=3. So S_3 = [2,3], A_3 = [1,1,2,2,3].
And so on.

Rémy Sigrist, Table of n, a(n) for n = 1..17711
Rémy Sigrist, PARI program for A342297

Cf. A000045, A000201, A001950, A181391.

PARI
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See Links section.
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A383447 Number of "peerless" trees on n nodes.

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A377912 Numbers such that every even digit except the last is immediately followed by a strictly larger digit.

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A376774 Indices of records in A376772.

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A376773 Records in A376772.

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A361337 Numbers that reach 0 after a suitable series of split-and-multiply operations (see Comments for precise definition).

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A361336 Smallest decimal number containing n palindromic substrings (Version 2). See Comments for precise definition.

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A361335 Smallest decimal number containing n palindromic substrings (Version 1). See Comments for precise definition.

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A361625 Number of free polyominoes with checkerboard-pattern-colored vertices with n cells.

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A361502 Index of n-th prime in A359804.

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