cp's OEIS Frontend

Number of permutations of degree n with at most one fall; called "Grassmannian permutations" by Lascoux and Schützenberger. - Axel Kohnert (Axel.Kohnert(AT)uni-bayreuth.de)

Number of different permutations of a deck of n cards that can be produced by a single shuffle. [DeSario]

Number of Dyck paths of semilength n having at most one long ascent (i.e., ascent of length at least two). Example: a(4)=12 because among the 14 Dyck paths of semilength 4, the only paths that have more than one long ascent are UUDDUUDD and UUDUUDDD (each with two long ascents). Here U = (1, 1) and D = (1, -1). Also number of ordered trees with n edges having at most one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004

Number of {12,1*2*,21*}-avoiding signed permutations in the hyperoctahedral group.

Number of 1342-avoiding circular permutations on [n+1].

2^n - n is the number of ways to partition {1, 2, ..., n} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths at least 1. - Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005

if b(0) = x and b(n) = b(n-1) + b(n-1)^2*x^(n-2) for n > 0, then b(n) is a polynomial of degree a(n). - Michael Somos, Nov 04 2006

The chromatic invariant of the Mobius ladder graph M_n for n >= 2. - Jonathan Vos Post, Aug 29 2008

Dimension sequence of the dual alternative operad (i.e., associative and satisfying the identity xyz + yxz + zxy + xzy + yzx + zyx = 0) over the field of characteristic 3. - Pasha Zusmanovich, Jun 09 2009

An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 26 and 176, lead to this sequence (without the first leading 1). For the central square these vectors lead to the companion sequence A168604. - Johannes W. Meijer, Aug 15 2010

a(n+1) is also the number of order-preserving and order-decreasing partial isometries (of an n-chain). - Abdullahi Umar, Jan 13 2011

A040001(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012

A130103(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012

The number of labeled graphs with n vertices whose vertex set can be partitioned into a clique and a set of isolated points. - Alex J. Best, Nov 20 2012

For n > 0, a(n) is a B_2 sequence. - Thomas Ordowski, Sep 23 2014

See coefficients of the linear terms of the polynomials of the table on p. 10 of the Getzler link. - Tom Copeland, Mar 24 2016

Consider n points lying on a circle, then for n>=2 a(n-1) is the maximum number of ways to connect two points with non-intersecting chords. - Anton Zakharov, Dec 31 2016

Also the number of cliques in the (n-1)-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

From Eric M. Schmidt, Jul 17 2017: (Start)

Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k). [Martinez and Savage, 2.7]

Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i), e(j), e(k) pairwise distinct. [Martinez and Savage, 2.7]

Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(j) >= e(k) and e(i) != e(k) pairwise distinct. [Martinez and Savage, 2.7]

(End)

Number of F-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are F-equivalent iff the positions of pattern F are identical in these paths. - Sergey Kirgizov, Apr 08 2018

From Gus Wiseman, Feb 10 2019: (Start)

Also the number of connected partitions of an n-cycle. For example, the a(1) = 1 through a(4) = 12 connected partitions are:

{{1}} {{12}} {{123}} {{1234}}

{{1}{2}} {{1}{23}} {{1}{234}}

{{12}{3}} {{12}{34}}

{{13}{2}} {{123}{4}}

{{1}{2}{3}} {{124}{3}}

{{134}{2}}

{{14}{23}}

{{1}{2}{34}}

{{1}{23}{4}}

{{12}{3}{4}}

{{14}{2}{3}}

{{1}{2}{3}{4}}

(End)

Number of subsets of n-set without the single-element subsets. - Yuchun Ji, Jul 16 2019

For every prime p, there are infinitely many terms of this sequence that are divisible by p (see IMO Compendium link and Doob reference). Corresponding indices n are: for p = 2, even numbers A299174; for p = 3, A047257; for p = 5, A349767. - Bernard Schott, Dec 10 2021

Primes are in A081296 and corresponding indices in A048744. - Bernard Schott, Dec 12 2021

First differs from A355744 at a(169) = 4, A355744(169) = 3.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Equivalently, numbers m such that 2^m - m is divisible by 3. Indeed, for every prime p, there are infinitely many numbers m such that 2^m - m (A000325) is divisible by p, here are numbers m corresponding to p = 3. - Bernard Schott, Dec 10 2021

Numbers k for which A276076(k) and A276086(k) are multiples of nine. For a simple proof, consider the penultimate digit in the factorial and primorial base expansions of n, A007623 and A049345. - Antti Karttunen, Feb 08 2024

A326123(n)/a(n) converges to 3/5.

a(n) is also the total area of the terraces of the first n even-indexed levels of the stepped pyramid described in A245092.

From Bernard Schott, Aug 06 2023: (Start)

Equivalently: positive numbers that are divisible by 3 or by 4.

Proof (similar to the proof proposed by Robert Israel in A355641).

If k is divisible by 3, then k is in the sequence because k = k/3 + k/3 + k/3.

If k is divisible by 4, then k is in the sequence because k = k/2 + k/4 + k/4.

Moreover, if k is positive and divisible by 6 (A008588), then k = k/3 + k/3 + k/3, but k is also in the sequence because k = k/2 + k/3 + k/6.

Conversely, to show that every term of this sequence is divisible by 3 or by 4, we consider all positive integer solutions of the equation 1 = 1/a + 1/b + 1/c. Without loss of generality, we may assume a <= b <= c, then 3/a >= 1/a + 1/b + 1/c = 1. So a <= 3. Similarly, given a, we have 2/b >= 1/b + 1/c = 1 - 1/a, so b <= 2/(1 - 1/a).

-> if a = 1, then 1 = 1 + 1/b + 1/c; this equation has clearly no solution.

-> if a = 2, then 1/2 = 1/b + 1/c with b <= 2/(1 - 1/2) = 4; in this case, there are two solutions: (a,b,c) = (2,3,6) or (a,b,c) = (2,4,4).

-> if a = 3, then 2/3 = 1/b + 1/c with b <= 2/(1 - 1/3) = 3; in this case, there is one solution: (a,b,c) = (3,3,3).

It turns out that there are only 3 solutions with a <= b <= c. Each corresponds to a possible pattern k = k/a + k/b + k/c for writing k as the sum of 3 of its divisors, which works when k is divisible by 3 or by 4. (End)

From David A. Corneth, Aug 07 2023: (Start)

Proof that a(n + 6) = a(n) + 12.

As k is in the sequence, k = k/d1 + k/d2 + k/d3 where d1, d2 and d3 | k and they are not necessarily distinct. By discussion above from Bernard Schott, Aug 06 2023, (d1, d2, d3) are in {(2, 3, 6), (2, 4, 4), (3, 3, 3)}. The lcm of these tuples are 6, 4 and 3 respectively. So any number k in the sequence is divisible by 3, 4 or 6.

This tells us that if k is in the sequence then k + 12 is in the sequence since k + 12 is divisible by one of 3, 4 or 6 since lcm(3, 4, 6) = 12.

So we can write a(n + m) = a(n) + 12 for some m. Inspection gives m = 6 so a(n + 6) = a(n) + 12. (End)

All terms are even. - Robert Israel, Aug 31 2022

Is it true that a(n) = 2*A080671(n)? - Michel Marcus, Sep 01 2022 (True for n <= 10000. - N. J. A. Sloane, Sep 01 2022)

This is true. In other words, k is in the sequence if and only if k is even and divisible by 3, 4 or 5. Proof: the positive integer solutions of 1/a + 1/b + 1/c + 1/d = 1 can be enumerated explicitly, and each contains at least one even number and at least one divisible by 3, 4 or 5. Of course k = k/a + k/b + k/c + k/d if and only if 1 = 1/a + 1/b + 1/c + 1/d, and this writes k as the sum of 4 divisors of k if k is divisible by a,b,c, and d. If k is even and divisible by 3, we can use 1 = 1/3 + 1/3 + 1/6 + 1/6; if divisible by 4, 1 = 1/4 + 1/4 + 1/4 + 1/4; if even and divisible by 5, 1 = 1/2 + 1/5 + 1/5 + 1/10. - Robert Israel, Sep 01 2022

The asymptotic density of this sequence is 11/30. - Amiram Eldar, Aug 08 2023

Numbers that are divisible by at least one of 5, 6, 8, 9, 14 and 21. For proof see link. - Robert Israel, Sep 01 2022

The asymptotic density of this sequence is 17/35. - Amiram Eldar, Aug 08 2023

Numbers divisible by at least one of 6, 8, 10, 14, 44, 52. For proof see link. - Robert Israel, Sep 02 2022

The asymptotic density of this sequence is 483/1430. - Amiram Eldar, Aug 08 2023

The sequence is well defined:

- a(1) = 3,

- for n > 0, let k be such that 2^k + 1 + a(1) + ... + a(n) < 2^(k+1),

- then a(n+1) <= 2^k + 1.

The variant where we avoid powers of 3 corresponds to the positive even numbers (A299174).