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

All terms except for the first are odd. - Joerg Arndt, Jul 19 2016

From Iain Fox, Nov 14 2017: (Start)

If k is congruent to 5 mod 6, then 3 divides 2^k - k; therefore a(n) is never congruent to 5 mod 6.

For even k, 2^k - k is divisible by 2; thus all terms other than 2 are odd.

It follows that for n > 1, a(n) is congruent to {1, 3} mod 6.

(End)

It is convenient, although not necessary, to let k be an odd number since k even => 2^k + k is even > 2.

Conjecture: sequence is infinite.

If k is prime we have A057664.

The next term is too big to display. Corresponding n are given in A058037.

The next term is too big to display. Corresponding n are given in A224470.

The next term is too big to display.

Corresponding n are given in A224471.

Corresponding values of m are given in A058046.

The next term has 254 digits.

Also semiprimes of the form 2^x - x.

The primes of the form 2^x - x are in A081296.

The next term is 2^735+735 = 18073..35103, 222 digits long.

a(6) > 1061095.

Intersection of A000040 and A048744.

Since numbers other than 3 that are congruent to 3 mod 6 are composite, for n > 2, a(n) is congruent to 1 mod 6 (see comments by Iain Fox in A048744).