cp's OEIS Frontend

This is also the result of applying the transformation on generating functions A -> 1/((1 - x)*(1 - x*A)) to the g.f. for the Catalan numbers.

p divides a(p) - 3 for prime p = 3 and p = {7, 13, 19, 31, 37, 43, ...} = A002476 (Primes of the form 6*n + 1). p^2 divides a(p^2) - 3 for prime p > 3. - Alexander Adamchuk, Jul 11 2006

Prime p divides a(p) for p = {2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ...} = A045309 (Primes congruent to {0, 2} mod 3); and A045309 (Primes p such that x^3 = n (integer) has only one solution mod p). Nonprime numbers n such that n divides a(n) are listed in A128287 = {1, 8, 133, ...}. - Alexander Adamchuk, Feb 23 2007

For p prime >= 5, a(p-1) = 1 or -2 (mod p) according as p = 1 or -1 (mod 3) (see Pan and Sun link). For example, with p=5, a(p-1) = 23 = -2 (mod p). - David Callan, Nov 29 2007

Hankel transform is A010892(n+1). - Paul Barry, Apr 24 2009

Equals INVERTi transform of A000245: (1, 3, 9, 28, ...). - Gary W. Adamson, May 15 2009

The subsequence of prime partial sums of Catalan numbers begins: a(1) = 2, a(4) = 23, a(6) = 197, a(16) = 48760367; see A121852. - Jonathan Vos Post, Feb 10 2010

Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k >= 1 including two kinds of (1,1). - Alois P. Heinz, Oct 14 2015

Binomial transform of A086246(n+1) = [1, 1, 1, 2, 4, 9, ...], or, equivalently, of A001006 (Motzkin numbers) with 1 prepended.

Or, n >= 0 appears 2^n times. - Jon Perry, Sep 21 2002

a(n) + 1 = number of bits in binary expansion of n.

Largest power of 2 dividing lcm(1..n): A007814(A003418(n)).

log_2(0) = -infinity.

Also Max_{k=1..n} Omega(k), where Omega(n) = A001222(n), number of prime factors with repetition; see A080613. - Reinhard Zumkeller, Feb 25 2003

From Paul Weisenhorn, Sep 29 2010, updated Aug 11 2020: (Start)

Arithmetic mean: m(1,(c+1)/c) = (2*c+1)/(2*c); harmonic mean: h(1,(c+1)/c) = 2*(c+1)/(2*c+1);

a(n) is the number of means to reach (n+1)/n from 2/1; with m for 0 and h for 1, the inverse binary expansion of n, without the leading 1, gives the sequence of means.

For example, n=20; inverse binary expansion without the leading 1: 0010 ---> m m h m or m(1, m(1, h(1, m(1, 2)))) = 21/20.

The 4 twofold means for n from 4 to 7:

m(1,m(1,2)) = m(1,3/2) = 5/4,

h(1,m(1,2)) = h(1,3/2) = 6/5,

m(1,h(1,2)) = m(1,4/3) = 7/6,

h(1,h(1,2)) = h(1,4/3) = 8/7. (End) [Edited by Petros Hadjicostas, Jul 23 2020]

As function of the absolute value, defines the minimal Euclidean function v on Z\{0}. A ring R is Euclidean if for some function v : R\{0}->N a division by nonzero b can be defined with remainder r satisfying either r=0 or v(r) < v(b). For the integers taking v(n)=|n| works, but v(n) = floor(log_2(|n|)) works as well; moreover it is the possibility with smallest possible values. For division by b>0 one can always choose |r| <= floor(b/2); this sequence satisfies a(1) = 0 and recursively a(n) = 1 + max(a(1), ..., a(floor(n/2))) for n > 1. - Marc A. A. van Leeuwen, Feb 16 2011

Maximum number of guesses required to find any k in a range of 1..n, with 'higher', 'lower' and 'correct' as answers. - Jon Perry, Nov 02 2013

Number of powers of 2 <= n. - Ralph-Joseph Tatt, Apr 23 2018

a(n) + 1 is the minimum number of pairwise disjoint subsets of an n-element set such that for each k from 1 to n there is a set with cardinality k which is the union of some of those subsets. - Wojciech Raszka, Apr 15 2019

Minimum height of an n-node binary tree. - Yuchun Ji, Mar 22 2021

If you are looking for the Schroeder numbers (a.k.a. large Schroder numbers, or big Schroeder numbers), see A006318.

Yang & Jiang (2021) call these the small 2-Schroeder numbers. - N. J. A. Sloane, Mar 28 2021

There are two schools of thought about the index for the first term. I prefer the indexing a(0) = a(1) = 1, a(2) = 3, a(3) = 11, etc.

a(n) is the number of ways to insert parentheses in a string of n+1 symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored.

Also length of list produced by a variant of the Catalan producing iteration: replace each integer k with the list 0,1,..,k,k+1,k,...,1,0 and get the length a(n) of the resulting (flattened) list after n iterations. - Wouter Meeussen, Nov 11 2001

Stanley gives several other interpretations for these numbers.

Number of Schroeder paths of semilength n (i.e., lattice paths from (0,0) to (2n,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no peaks at level 1. Example: a(2)=3 because among the six Schroeder paths of semilength two HH, UHD, UUDD, HUD, UDH and UDUD, only the first three have no peaks at level 1. - Emeric Deutsch, Dec 27 2003

a(n+1) is the number of Dyck n-paths in which the interior vertices of the ascents are colored white or black. - David Callan, Mar 14 2004

Number of possible schedules for n time slots in the first-come first-served (FCFS) printer policy.

Also row sums of A086810, A033282, A126216. - Philippe Deléham, May 09 2004

a(n+1) is the number of pairs (u,v) of same-length compositions of n, 0's allowed in u but not in v and u dominates v (meaning u_1 >= v_1, u_1 + u_2 >= v_1 + v_2 and so on). For example, with n=2, a(3) counts (2,2), (1+1,1+1), (2+0,1+1). - David Callan, Jul 20 2005

The big Schroeder number (A006318) is the number of Schroeder paths from (0,0) to (n,n) (subdiagonal paths with steps (1,0) (0,1) and (1,1)). These paths fall in two classes: those with steps on the main diagonal and those without. These two classes are equinumerous and the number of paths in either class is the little Schroeder number a(n) (half the big Schroeder number). - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005

With offset 0, a(n) = number of (colored) Motzkin (n-1)-paths with each upstep U getting one of 2 colors and each flatstep F getting one of 3 colors. Example. With their colors immediately following upsteps/flatsteps, a(2) = 3 counts F1, F2, F3 and a(3)=11 counts U1D, U2D, F1F1, F1F2, F1F3, F2F1, F2F2, F2F3, F3F1, F3F2, F3F3. - David Callan, Aug 16 2006

Shifts left when INVERT transform applied twice. - Alois P. Heinz, Apr 01 2009

Number of increasing tableaux of shape (n,n). An increasing tableau is a semistandard tableaux with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 3 because of the three tableaux (12)(34), (13)(24), (12)(23). - Oliver Pechenik, Apr 22 2014

Number of ordered trees with no vertex of outdegree 1 and having n+1 leaves (called sometimes Schröder trees). - Emeric Deutsch, Dec 13 2014

Number of dissections of a convex (n+2)-gon by nonintersecting diagonals. Example: a(2)=3 because for a square ABCD we have (i) no diagonal, (ii) dissection with diagonal AC, and (iii) dissection with diagonal BD. - Emeric Deutsch, Dec 13 2014

The little Schroeder numbers are the moments of the Marchenko-Pastur law for the case c=2 (although the moment m0 is 1/2 instead of 1): 1/2, 1, 3, 11, 45, 197, 903, ... - Jose-Javier Martinez, Apr 07 2015

Number of generalized Motzkin paths with no level steps at height 0, from (0,0) to (2n,0), and consisting of steps U=(1,1), D=(1,-1) and H2=(2,0). For example, for n=3, we have the 11 paths: UDUDUD, UUDDUD, UDUUDD, UH2DUD, UDUH2D, UH2H2D, UUDUDD, UUUDDD, UUH2DD, UUDH2D, UH2UDD. - José Luis Ramírez Ramírez, Apr 20 2015

REVERT transform of A225883. - Vladimir Reshetnikov, Oct 25 2015

Total number of (nonempty) faces of all dimensions in the associahedron K_{n+1} of dimension n-1. For example, K_4 (a pentagon) includes 5 vertices and 5 edges together with K_4 itself (5 + 5 + 1 = 11), while K_5 includes 14 vertices, 21 edges and 9 faces together with K_5 itself (14 + 21 + 9 + 1 = 45). - Noam Zeilberger, Sep 17 2018

a(n) is the number of interval posets of permutations with n minimal elements that have exactly two realizers, up to a shift by 1 in a(4). See M. Bouvel, L. Cioni, B. Izart, Theorem 17 page 13. - Mathilde Bouvel, Oct 21 2021

a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_1 = 1, (ii) u_i <= i for all i, (iii) the nonzero u_i are weakly increasing. For example, a(2) = 3 counts 10, 11, 12, and a(3) = 11 counts 100, 101, 102, 103, 110, 111, 112, 113, 120, 122, 123. See link below. - David Callan, Dec 19 2021

a(n) is the number of parking functions of size n avoiding the patterns 132 and 213. - Lara Pudwell, Apr 10 2023

a(n+1) is the number of Schroeder paths from (0,0) to (2n,0) in which level steps at height 0 come in 2 colors. - Alexander Burstein, Jul 23 2023

a(n) is also the size of the automorphism group of the graph (edge graph) of the n-dimensional hypercube and also of the geometric automorphism group of the hypercube (the two groups are isomorphic). This group is an extension of an elementary Abelian group (C_2)^n by S_n. (C_2 is the cyclic group with two elements and S_n is the symmetric group.) - Avi Peretz (njk(AT)netvision.net.il), Feb 21 2001

Then a(n) appears in the power series: sqrt(1+sin(y)) = Sum_{n>=0} (-1)^floor(n/2)*y^(n)/a(n) and sqrt((1+cos(y))/2) = Sum_{n>=0} (-1)^n*y^(2n)/a(2n). - Benoit Cloitre, Feb 02 2002

Appears to be the BinomialMean transform of A001907. See A075271. - John W. Layman, Sep 28 2002

Number of n X n monomial matrices with entries 0, +-1.

Also number of linear signed orders.

Define a "downgrade" to be the permutation d which places the items of a permutation p in descending order. This note concerns those permutations that are equal to their double-downgrades. The number of permutations of order 2n having this property are equinumerous with those of order 2n+1. a(n) = number of double-downgrading permutations of order 2n and 2n+1. - Eugene McDonnell (eemcd(AT)mac.com), Oct 27 2003

a(n) = (Integral_{x=0..Pi/2} cos(x)^(2*n+1) dx) where the denominators are b(n) = (2*n)!/(n!*2^n). - Al Hakanson (hawkuu(AT)excite.com), Mar 02 2004

1 + (1/2)x - (1/8)x^2 - (1/48)x^3 + (1/384)x^4 + ... = sqrt(1+sin(x)).

a(n)*(-1)^n = coefficient of the leading term of the (n+1)-th derivative of arctan(x), see Hildebrand link. - Reinhard Zumkeller, Jan 14 2006

a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is j for iDavid Callan, Sep 25 2006

a(n) is the number of increasing plane trees with n+1 edges. (In a plane tree, each subtree of the root is an ordered tree but the subtrees of the root may be cyclically rotated.) Increasing means the vertices are labeled 0,1,2,...,n+1 and each child has a greater label than its parent. Cf. A001147 for increasing ordered trees, A000142 for increasing unordered trees and A000111 for increasing 0-1-2 trees. - David Callan, Dec 22 2006

Hamed Hatami and Pooya Hatami prove that this is an upper bound on the cardinality of any minimal dominating set in C_{2n+1}^n, the Cartesian product of n copies of the cycle of size 2n+1, where 2n+1 is a prime. - Jonathan Vos Post, Jan 03 2007

This sequence and (1,-2,0,0,0,0,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland, Oct 29 2007

a(n) = number of permutations of the multiset {1,1,2,2,...,n,n,n+1,n+1} such that between the two occurrences of i, there is exactly one entry >i, for i=1,2,...,n. Example: a(2) = 8 counts 121323, 131232, 213123, 231213, 232131, 312132, 321312, 323121. Proof: There is always exactly one entry between the two 1s (when n>=1). Given a permutation p in A(n) (counted by a(n)), record the position i of the first 1, then delete both 1s and subtract 1 from every entry to get a permutation q in A(n-1). The mapping p -> (i,q) is a bijection from A(n) to the Cartesian product [1,2n] X A(n-1). - David Callan, Nov 29 2007

Row sums of A028338. - Paul Barry, Feb 07 2009

a(n) is the number of ways to seat n married couples in a row so that everyone is next to their spouse. Compare A007060. - Geoffrey Critzer, Mar 29 2009

From Gary W. Adamson, Apr 21 2009: (Start)

Equals (-1)^n * (1, 1, 2, 8, 48, ...) dot (1, -3, 5, -7, 9, ...).

Example: a(4) = 384 = (1, 1, 2, 8, 48) dot (1, -3, 5, -7, 9) = (1, -3, 10, -56, 432). (End)

exp(x/2) = Sum_{n>=0} x^n/a(n). - Jaume Oliver Lafont, Sep 07 2009

Assuming n starts at 0, a(n) appears to be the number of Gray codes on n bits. It certainly is the number of Gray codes on n bits isomorphic to the canonical one. Proof: There are 2^n different starting positions for each code. Also, each code has a particular pattern of bit positions that are flipped (for instance, 1 2 1 3 1 2 1 for n=3), and these bit position patterns can be permuted in n! ways. - D. J. Schreffler (ds1404(AT)txstate.edu), Jul 18 2010

E.g.f. of 0,1,2,8,... is x/(1-2x/(2-2x/(3-8x/(4-8x/(5-18x/(6-18x/(7-... (continued fraction). - Paul Barry, Jan 17 2011

Number of increasing 2-colored trees with choice of two colors for each edge. In general, if we replace 2 with k we get the number of increasing k-colored trees. For example, for k=3 we get the triple factorial numbers. - Wenjin Woan, May 31 2011

a(n) = row sums of triangle A193229. - Gary W. Adamson, Jul 18 2011

Also the number of permutations of 2n (or of 2n+1) that are equal to their reverse-complements. (See the Egge reference.) Note that the double-downgrade described in the preceding comment (McDonnell) is equivalent to the reverse-complement. - Justin M. Troyka, Aug 11 2011

The e.g.f. can be used to form a generator, [1/(1-2x)] d/dx, for A000108, so a(n) can be applied to A145271 to generate the Catalan numbers. - Tom Copeland, Oct 01 2011

The e.g.f. of 1/a(n) is BesselI(0,sqrt(2*x)). See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - Wolfdieter Lang, Jan 09 2012

a(n) = order of the largest imprimitive group of degree 2n with n systems of imprimitivity (see [Miller], p. 203). - L. Edson Jeffery, Feb 05 2012

Row sums of triangle A208057. - Gary W. Adamson, Feb 22 2012

a(n) is the number of ways to designate a subset of elements in each n-permutation. a(n) = A000142(n) + A001563(n) + A001804(n) + A001805(n) + A001806(n) + A001807(n) + A035038(n) * n!. - Geoffrey Critzer, Nov 08 2012

For n>1, a(n) is the order of the Coxeter groups (also called Weyl groups) of types B_n and C_n. - Tom Edgar, Nov 05 2013

For m>0, k*a(m-1) is the m-th cumulant of the chi-squared probability distribution for k degrees of freedom. - Stanislav Sykora, Jun 27 2014

a(n) with 0 prepended is the binomial transform of A120765. - Vladimir Reshetnikov, Oct 28 2015

Exponential self-convolution of A001147. - Vladimir Reshetnikov, Oct 08 2016

Also the order of the automorphism group of the n-ladder rung graph. - Eric W. Weisstein, Jul 22 2017

a(n) is the order of the group O_n(Z) = {A in M_n(Z): A*A^T = I_n}, the group of n X n orthogonal matrices over the integers. - Jianing Song, Mar 29 2021

a(n) is the number of ways to tile a (3n,3n)-benzel or a (3n+1,3n+2)-benzel using left stones and two kinds of bones; see Defant et al., below. - James Propp, Jul 22 2023

a(n) is the number of labeled histories for a labeled topology with the modified lodgepole shape and n+1 cherry nodes. - Noah A Rosenberg, Jan 16 2025

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(12).

Binary expansion ends 11.

These the numbers for which zeta(2*x+1) needs just 2 terms to be evaluated. - Jorge Coveiro, Dec 16 2004 [This comment needs clarification]

a(n) is the smallest k such that for every r from 0 to 2n - 1 there exist j and i, k >= j > i > 2n - 1, such that j - i == r (mod (2n - 1)), with (k, (2n - 1)) = (j,(2n - 1)) = (i, (2n - 1)) = 1. - Amarnath Murthy, Sep 24 2003

Complement of A004773. - Reinhard Zumkeller, Aug 29 2005

Any (4n+3)-dimensional manifold endowed with a mixed 3-Sasakian structure is an Einstein space with Einstein constant lambda = 4n + 2 [Theorem 3, p. 10 of Ianus et al.]. - Jonathan Vos Post, Nov 24 2008

Solutions to the equation x^(2*x) = 3*x (mod 4*x). - Farideh Firoozbakht, May 02 2010

Subsequence of A022544. - Vincenzo Librandi, Nov 20 2010

First differences of A084849. - Reinhard Zumkeller, Apr 02 2011

Numbers n such that {1, 2, 3, ..., n} is a losing position in the game of Nim. - Franklin T. Adams-Watters, Jul 16 2011

Numbers n such that there are no primes p that satisfy the relationship p XOR n = p + n. - Brad Clardy, Jul 22 2012

The XOR of all numbers from 1 to a(n) is 0. - David W. Wilson, Apr 21 2013

A089911(4*a(n)) = 4. - Reinhard Zumkeller, Jul 05 2013

First differences of A014105. - Ivan N. Ianakiev, Sep 21 2013

All triangular numbers in the sequence are congruent to {3, 7} mod 8. - Ivan N. Ianakiev, Nov 12 2013

Apart from the initial term, length of minimal path on an n-dimensional cubic lattice (n > 1) of side length 2, until a self-avoiding walk gets stuck. Construct a path connecting all 2n points orthogonally adjacent from the center, ending at the center. Starting at any point adjacent to the center, there are 2 steps to reach each of the remaining 2n - 1 points, resulting in path length 4n - 2 with a final step connecting the center, for a total path length of 4n - 1, comprising 4n points. - Matthew Lehman, Dec 10 2013

a(n-1), n >= 1, appears as first column in the triangles A238476 and A239126 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014

For the Collatz Conjecture, we identify two types of odd numbers. This sequence contains all the ascenders: where (3*a(n) + 1) / 2 is odd and greater than a(n). See A016813 for the descenders. - Jaroslav Krizek, Jul 29 2016

Note that when starting from a(n)^2, equality holds between series of first n+1 and next n consecutive squares: a(n)^2 + (a(n) + 1)^2 + ... + (a(n) + n)^2 = (a(n) + n + 1)^2 + (a(n) + n + 2)^2 + ... + (a(n) + 2*n)^2; e.g., 10^2 + 11^2 + 12^2 = 13^2 + 14^2. - Henry Bottomley, Jan 22 2001; with typos fixed by Zak Seidov, Sep 10 2015

a(n) = sum of second set of n consecutive even numbers - sum of the first set of n consecutive odd numbers: a(1) = 4-1, a(3) = (8+10+12) - (1+3+5) = 21. - Amarnath Murthy, Nov 07 2002

Partial sums of odd numbers 3 mod 4, that is, 3, 3+7, 3+7+11, ... See A001107. - Jon Perry, Dec 18 2004

If Y is a fixed 3-subset of a (2n+1)-set X then a(n) is the number of (2n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

More generally (see the first comment), for n > 0, let b(n,k) = a(n) + k*(4*n + 1). Then b(n,k)^2 + (b(n,k) + 1)^2 + ... + (b(n,k) + n)^2 = (b(n,k) + n + 1 + 2*k)^2 + ... + (b(n,k) + 2*n + 2*k)^2 + k^2; e.g., if n = 3 and k = 2, then b(n,k) = 47 and 47^2 + ... + 50^2 = 55^2 + ... + 57^2 + 2^2. - Charlie Marion, Jan 01 2011

Sequence found by reading the line from 0, in the direction 0, 10, ..., and the line from 3, in the direction 3, 21, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Nov 09 2011

a(n) is the number of positions of a domino in a pyramidal board with base 2n+1. - César Eliud Lozada, Sep 26 2012

Differences of row sums of two consecutive rows of triangle A120070, i.e., first differences of A016061. - J. M. Bergot, Jun 14 2013 [In other words, the partial sums of this sequence give A016061. - Leo Tavares, Nov 23 2021]

a(n)*Pi is the total length of half circle spiral after n rotations. See illustration in links. - Kival Ngaokrajang, Nov 05 2013

For corresponding sums in first comment by Henry Bottomley, see A059255. - Zak Seidov, Sep 10 2015

a(n) also gives the dimension of the simple Lie algebras B_n (n >= 2) and C_n (n >= 3). - Wolfdieter Lang, Oct 21 2015

With T_(i+1,i)=a(i+1) and all other elements of the lower triangular matrix T zero, T is the infinitesimal generator for unsigned A130757, analogous to A132440 for the Pascal matrix. - Tom Copeland, Dec 13 2015

Partial sums of squares with alternating signs, ending in an even term: a(n) = 0^2 - 1^2 +- ... + (2*n)^2, cf. Example & Formula from Berselli, 2013. - M. F. Hasler, Jul 03 2018

Also numbers k with the property that in the symmetric representation of sigma(k) the smallest Dyck path has a central peak and the largest Dyck path has a central valley, n > 0. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

a(n) is the area of a triangle with vertices at (0,0), (2*n+1, 2*n), and ((2*n+1)^2, 4*n^2). - Art Baker, Dec 12 2018

This sequence is the largest subsequence of A000217 such that gcd(a(n), 2*n) = a(n) mod (2*n) = n, n > 0 up to a given value of n. It is the interleave of A033585 (a(n) is even) and A033567 (a(n) is odd). - Torlach Rush, Sep 09 2019

A generalization of Hasler's Comment (Jul 03 2018) follows. Let P(k,n) be the n-th k-gonal number. Then for k > 1, partial sums of {P(k,n)} with alternating signs, ending in an even term, = n*((k-2)*n + 1). - Charlie Marion, Mar 02 2021

Let U_n(H) = {A in M_n(H): A*A^H = I_n} be the group of n X n unitary matrices over the quaternions (A^H is the conjugate transpose of A. Note that over the quaternions we still have A*A^H = I_n <=> A^H*A = I_n by mapping A and A^H to (2n) X (2n) complex matrices), then a(n) is the dimension of its Lie algebra u_n(H) = {A in M_n(H): A + A^H = 0} as a real vector space. A basis is given by {(E_{st}-E_{ts}), i*(E_{st}+E_{ts}), j*(E_{st}+E_{ts}), k*(E_{st}+E_{ts}): 1 <= s < t <= n} U {i*E_{tt}, j*E_{tt}, k*E_{tt}: t = 1..n}, where E_{st} is the matrix with all entries zero except that its (st)-entry is 1. - Jianing Song, Apr 05 2021

Number of peaks at even level in all Dyck paths of semilength n+1. Example: a(2)=4 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUUDDD, where U=(1,1), D=(1,-1) and the peaks at even level are shown by *. - Emeric Deutsch, Dec 05 2003

Also number of long ascents (i.e., ascents of length at least two) in all Dyck paths of semilength n+1. Example: a(2)=4 because in the five Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and (UUU)DDD, we have four long ascents (shown between parentheses). Here U=(1,1) and D=(1,-1). Also number of branch nodes (i.e., vertices of outdegree at least two) in all ordered trees with n+1 edges. - Emeric Deutsch, Feb 22 2004

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=1. Example: For n=2 these are the paths EENN, ENEN, ENNE and NEEN. - Herbert Kociemba, May 23 2004

Narayana transform (A001263) of [1, 3, 5, 7, 9, ...] = (1, 4, 15, 56, 210, ...). Row sums of triangles A136534 and A136536. - Gary W. Adamson, Jan 04 2008

Starting with offset 1 = the Catalan sequence starting (1, 2, 5, 14, ...) convolved with A000984: (1, 2, 6, 20, ...). - Gary W. Adamson, May 17 2009

Also number of peaks plus number of valleys in all Dyck n-paths. - David Scambler, Oct 08 2012

Apparently counts UDDUD in all Dyck paths of semilength n+2. - David Scambler, Apr 22 2013

Apparently the number of peaks strictly left of the midpoint in all Dyck paths of semilength n+1. - David Scambler, Apr 30 2013

For n>0, a(n) is the number of compositions of n into at most n parts if zeros are allowed as parts (so-called "weak" compositions). - L. Edson Jeffery, Jul 24 2014

Number of paths in the half-plane x >= 0, from (0,0) to (2n,2), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=2, we have the 4 paths: UUUD, UUDU, UDUU, DUUU. - José Luis Ramírez Ramírez, Apr 19 2015

For n>1, 1/a(n) is the probability that when a stick is broken up at n points independently and uniformly chosen at random along its length any triple of pieces of the n+1 pieces can form a triangle. The corresponding probability for the existence of at least one triple is A339392(n)/A339393(n). - Amiram Eldar, Dec 04 2020

a(n) is the number of lattice paths of 2n steps taken from the step set {U=(1,1), D=(1,-1)} that start at the origin, never go below the x-axis, and end strictly above the x-axis; more succinctly, proper left factors of Dyck paths. For example, a(2)=4 counts UUUU, UUUD, UUDU, UDUU. - David Callan and Emeric Deutsch, Jan 25 2021

From Gus Wiseman, Jul 21 2021: (Start)

Also the number of integer compositions of 2n+1 with alternating sum -1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(1) = 1 through a(3) = 15 compositions are:

(1,2) (2,3) (3,4)

(1,3,1) (1,4,2)

(1,1,1,2) (2,4,1)

(1,2,1,1) (1,1,2,3)

(1,2,2,2)

(1,3,2,1)

(2,1,1,3)

(2,2,1,2)

(2,3,1,1)

(1,1,1,3,1)

(1,2,1,2,1)

(1,3,1,1,1)

(1,1,1,1,1,2)

(1,1,1,2,1,1)

(1,2,1,1,1,1)

The following relate to these compositions.

- The unordered version is A000070.

- Allowing any negative alternating sum gives A000346.

- The opposite (positive 1) version is A000984.

- The version for reverse-alternating sum is also A001791 (this sequence).

- Taking alternating sum -2 instead of -1 gives A002054.

- The shifted version for alternating sum 0 is counted by A088218 and ranked by A344619.

- Ranked by A345910 (reverse: A345912).

Equivalently, a(n) counts binary numbers with 2n+1 digits and one more 0 than 1's. For example, the a(2) = 4 binary numbers are: 10001, 10010, 10100, 11000.

(End)

The diagonal of a square n X n matrix where cells of the first row are the nonnegative integers and cells of subsequent rows are sums of cells of the previous row up to and including n. - Torlach Rush, Apr 24 2024

For n>=1, a(n) is the independence number of the odd graph O_{n+1}. - Miquel A. Fiol, Jul 07 2024

These are Hogben's central polygonal numbers

2

.P

3 n

Also the sum of three consecutive triangular numbers (A000217); i.e., a(4) = 19 = T4 + T3 + T2 = 10 + 6 + 3. - Robert G. Wilson v, Apr 27 2001

For k>2, Sum_{n=1..k} a(n) gives the sum pertaining to the magic square of order k. E.g., Sum_{n=1..5} a(n) = 1 + 4 + 10 + 19 + 31 = 65. In general, Sum_{n=1..k} a(n) = k*(k^2 + 1)/2. - Amarnath Murthy, Dec 22 2001

Binomial transform of (1,3,3,0,0,0,...). - Paul Barry, Jul 01 2003

a(n) is the difference of two tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-2)(n-1)(n)/6. - Alexander Adamchuk, May 20 2006

Partial sums are A006003(n) = n(n^2+1)/2. Finite differences are a(n+1) - a(n) = A008585(n) = 3n. - Alexander Adamchuk, Jun 03 2006

If X is an n-set and Y a fixed 3-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

Equals (1, 2, 3, ...) convolved with (1, 2, 3, 3, 3, ...). a(4) = 19 = (1, 2, 3, 4) dot (3, 3, 2, 1) = (3 + 6 + 6 + 4). - Gary W. Adamson, May 01 2009

Equals the triangular numbers convolved with [1, 1, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009

a(n) is the number of triples (w,x,y) having all terms in {0,...,n} and min(w+x,x+y,y+w) = max(w,x,y). - Clark Kimberling, Jun 14 2012

a(n) = number of atoms at graph distance <= n from an atom in the graphite or graphene network (cf. A008486). - N. J. A. Sloane, Jan 06 2013

In 1826, Shiraishi gave a solution to the Diophantine equation a^3 + b^3 + c^3 = d^3 with b = a(n) for n > 1; see A226903. - Jonathan Sondow, Jun 22 2013

For n > 1, a(n) is the remainder of n^2 * (n-1)^2 mod (n^2 + (n-1)^2). - J. M. Bergot, Jun 27 2013

The equation A000578(x) - A000578(x-1) = A000217(y) - A000217(y-2) is satisfied by y=a(x). - Bruno Berselli, Feb 19 2014

A242357(a(n)) = n. - Reinhard Zumkeller, May 11 2014

A255437(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2015

The first differences give A008486. a(n) seems to give the total number of triangles in the n-th generation of the six patterns of triangle expansion shown in the link. - Kival Ngaokrajang, Sep 12 2015

Number of binary shuffle squares of length 2n which contains exactly two 1's. - Bartlomiej Pawlik, Sep 07 2023

The digital root has period 3 (1, 4, 1) (A146325), the same digital root as the centered 12-gonal numbers, or centered dodecagonal numbers A003154(n). - Peter M. Chema, Dec 20 2023

This entry is a list, and so has offset 1. WARNING: However, in this entry several comments, formulas and programs seem to refer to the original version of this sequence which had offset 0. - M. F. Hasler, Oct 06 2014

Number of necklaces with n-1 beads and two colors that are the same when turned over and hence have reflection symmetry. [edited by Herbert Kociemba, Nov 24 2016]

The subset {a(1),...,a(2k)} contains all proper divisors of 3*2^k. - Ralf Stephan, Jun 02 2003

Let k = any nonnegative integer and j = 0 or 1. Then n+1 = 2k + 3j and a(n) = 2^k*3^j. - Andras Erszegi (erszegi.andras(AT)chello.hu), Jul 30 2005

Smallest number having no fewer prime factors than any predecessor, a(0)=1; A110654(n) = A001222(a(n)); complement of A116451. - Reinhard Zumkeller, Feb 16 2006

A093873(a(n)) = 1. - Reinhard Zumkeller, Oct 13 2006

a(n) = a(n-1) + a(n-2) - gcd(a(n-1), a(n-2)), n >= 3, a(1)=2, a(2)=3. - Ctibor O. Zizka, Jun 06 2009

Where records occur in A048985: A193652(n) = A048985(a(n)) and A193652(n) < A048985(m) for m < a(n). - Reinhard Zumkeller, Aug 08 2011

A002348(a(n)) = A000079(n-3) for n > 2. - Reinhard Zumkeller, Mar 18 2012

Without initial 1, third row in array A228405. - Richard R. Forberg, Sep 06 2013

Also positions of records in A048673. A246360 gives the record values. - Antti Karttunen, Sep 23 2014

Known in numerical mathematics as "Bulirsch sequence", used in various extrapolation methods for step size control. - Peter Luschny, Oct 30 2019

For n > 1, squares of the terms can be expressed as the sum of two powers of two: 2^x + 2^y. - Karl-Heinz Hofmann, Sep 08 2022

This sequence represents the expected saturation of a binary search tree (or BST) on n nodes times the number of binary search trees on n nodes, or alternatively, the sum of the saturation of all binary search trees on n nodes. - Marko Riedel, Jan 24 2002

1->12, 2->123, 3->1234 etc. starting with 1, gives A007001: 1, 12, 12123, 12123121231234... summing the digits gives this sequence. - Miklos Kristof, Nov 05 2002

a(n-1) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003

With offset 1, number of permutations beginning with 12 and avoiding 32-1.

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=1. - Herbert Kociemba, May 24 2004

a(n) is the number of Dyck (n+1)-paths that start with UU. For example, a(2)=3 counts UUUDDD, UUDUDD, UUDDUD. - David Callan, Dec 08 2004

a(n) is the number of Dyck (n+2)-paths that start with UUDU. For example, a(2)=3 counts UUDUDDUD, UUDUDUDD, UUDUUDDD. - David Scambler, Feb 14 2011

Hankel transform is (0,-1,-1,0,1,1,0,-1,-1,0,...). Hankel transform of a(n+1) is (1,0,-1,-1,0,1,1,0,-1,-1,0,...). - Paul Barry, Feb 08 2008

Starting with offset 1 = row sums of triangle A154558. - Gary W. Adamson, Jan 11 2009

Starting with offset 1 equals INVERT transform of A014137, partial sums of the Catalan numbers: (1, 2, 4, 9, 23, ...). - Gary W. Adamson, May 15 2009

With offset 1, a(n) is the binomial transform of the shortened Motzkin numbers: 1, 2, 4, 9, 21, 51, 127, 323, ... (A001006). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Sep 07 2009

The Catalan sequence convolved with its shifted variant, e.g. a(5) = 90 = (1, 1, 2, 5, 14) dot (42, 14, 5, 2, 1) = (42 + 14 + 10 + 10 + 14 ) = 90. - Gary W. Adamson, Nov 22 2011

a(n+2) = A214292(2*n+3,n). - Reinhard Zumkeller, Jul 12 2012

With offset 3, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding, i.e., do not contain a three term monotone subsequence, for which the first ascent is at positions (3,4); see Connolly link. There it is shown in general that the k-th Catalan Convolution is the number of 123-avoiding permutations for which the first ascent is at (k, k+1). (For n=k, the first ascent is defined to be at positions (k,k+1) if the permutation is the decreasing permutation with no ascents.) - Anant Godbole, Jan 17 2014

With offset 3, a(n)=number of permutations on {1,2,...,n} that are 123-avoiding and for which the integer n is in the 3rd spot; see Connolly link. For example, there are 297 123-avoiding permutations on n=9 at which the element 9 is in the third spot. - Anant Godbole, Jan 17 2014

With offset 1, a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off y = x to the right exactly once but do not cross y = x vertically. Details can be found in Section 4.4 in Pan and Remmel's link. - Ran Pan, Feb 01 2016

The total number of returns (downsteps which end on the line y=0) within the set of all Dyck paths from (0,0) to (2n,0). - Cheyne Homberger, Sep 05 2016

a(n) is the number of intervals of the form [s,w] that are distributive (equivalently, modular) lattices in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020

a(n+2) is the number of coalescent histories for a pair (G,S) where G is a gene tree with 3-pseudocaterpillar shape and n leaves, S is a species tree with caterpillar shape and n leaves, and G and S have identical leaf labeling. - Noah A Rosenberg, Jun 15 2022

a(n) is the number of parking functions of size n avoiding the patterns 132, 213, and 312. - Lara Pudwell, Apr 10 2023

a(n) is the number of parking functions of size n avoiding the patterns 123 and 213. - Lara Pudwell, Apr 10 2023