cp's OEIS Frontend

Equivalently, walks on triangle, visiting n+2 vertices, so length n+1, n "corners"; the symmetry group is S3, reversing a walk does not count as different. Walks are not self-avoiding. - Colin Mallows

Slavik V. Jablan observes that this is also the number of rational knots and links with n+2 crossings (cf. A018240). See reference. [Corrected by Andrey Zabolotskiy, Jun 18 2020]

Number of bit strings of length (n-1), not counting strings which are the end-for-end reversal or the 0-for-1 reversal of each other as different. - Carl Witty (cwitty(AT)newtonlabs.com), Oct 27 2001

The formula given in page 1095 of the Balasubramanian reference can be used to derive this sequence. - Parthasarathy Nambi, May 14 2007

Also number of compositions of n up to direction, where a composition is considered equivalent to its reversal, see example. - Franklin T. Adams-Watters, Oct 24 2009

Number of normally non-isomorphic realizations of the associahedron of type I starting with dimension 2 in Ceballos et al. - Tom Copeland, Oct 19 2011

Number of fibonacenes with n+2 hexagons. See the Balaban and the Dobrynin references. - Emeric Deutsch, Apr 21 2013

From the point of view of binary grids, it is a (1,n)-rectangular grid. A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11. - Yosu Yurramendi, May 19 2013

Number of n-vertex difference graphs (bipartite 2K_2-free graphs) [Peled & Sun, Thm. 9]. - Falk Hüffner, Jan 10 2016

The offset should be 0, since the first row of A034851 is row 0. The name would then be: "Number of n bead...". - Daniel Forgues, Jul 26 2018

a(n) is the number of non-isomorphic generalized rigid ladders with n cells. A generalized rigid ladder with n cells is a graph with vertex set is the union of {u_0, u_1, ..., u_n} and {v_0, v_1, ..., v_n}, and for every 0 <= i <= n-1, the edges are of the form {u_i,u_i+1}, {v_i, v_i+1}, {u_i,v_i} and either {u_i,v_i+1} or {u_i+1,v_i}. - Christian Barrientos, Jul 29 2018

Also number of non-isomorphic stairs with n+1 cells. A stair is a snake polyomino allowing only two directions for adjacent cells: east and north. - Christian Barrientos and Sarah Minion, Jul 29 2018

From Robert A. Russell, Oct 28 2018: (Start)

There are two different unoriented row colorings using two colors that give us very similar results here, a difference of one in the offset. In an unoriented row, chiral pairs are counted as one.

a(n) is the number of color patterns (set partitions) of an unoriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if the colors are permutable.

a(n+1) is the number of ways to color an unoriented row of length n using two noninterchangeable colors (one need not use both colors).

See the examples below of these two different colorings. (End)

Also arises from the enumeration of types of based polyhedra with exactly two triangular faces [Rademacher]. - N. J. A. Sloane, Apr 24 2020

a(n) is the number of (unlabeled) 2-paths with n+4 vertices. (A 2-path with order n at least 4 can be constructed from a 3-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to an existing 2-clique containing an existing 2-leaf.) - Allan Bickle, Apr 05 2022

a(n) is the number of caterpillars with a perfect matching and order 2n+2. - Christian Barrientos, Sep 12 2023

a(n) is also the number of distinct planar embeddings of the (n+2)-centipede graph (up to at least n=8 and likely for all larger n). - Eric W. Weisstein, May 21 2024

a(n) is also the number of distinct planar embeddings of the 2 X (n+2) grid graph i.e., the (n+2)-ladder graph. - Eric W. Weisstein, May 21 2024

Dimension of the homogeneous component of degree n of the free Jordan algebra on two generators (or, in this case, the free special Jordan algebra on two generators). It follows from (Shirshov 1956, Cohn 1959). - Vladimir Dotsenko, Mar 29 2025

Sum of first n^2 positive integers.

Start from xanthene and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink below on chemistry. - Robert G. Wilson v, Aug 02 2002; Amarnath Murthy, Aug 01 2002

Sum of the next n multiples of n. - Amarnath Murthy, Aug 01 2002

The sum of the terms in an n X n spiral. These are also triangular numbers. - William A. Tedeschi, Feb 27 2008

Hypotenuse of Pythagorean triangles with smallest side a cube: A000578(n)^2 + A083374(n)^2 = a(n)^2. - Martin Renner, Nov 12 2011

For n>1, triangular numbers that can be represented as a sum of a square and a triangular number. For example, a(2)=10=4+6=9+1. - Ivan N. Ianakiev, Apr 24 2012

A037270 can be constructed in the following manner: Take A000217 and for every n not in A000290 delete the corresponding A000217(n). - Ivan N. Ianakiev, Apr 26 2012

Starting at a(1)=1 simply take 1*1=1, a(2)= 2*(2+3)=10, a(3)= 3*(4+5+6)=45, a(4)=4*(7+8+9+10) and so on. - J. M. Bergot, May 01 2015

Observation: The digital roots of the terms repeat in the sequence 1, 1, 9; e.g., the digital roots of 1, 10, 45, 136, 325, and 666 are 1, 1, 9, 1, 1, and 9. Verified for the first 10000 terms. - Rob Barton, Mar 28 2018

The above observation is easily explained and proved given that the digital root of a positive number equals the number modulo 9, and a(n + 9k) == a(n) (mod 9). - M. F. Hasler, Apr 05 2018

Number of unoriented rows of length 4 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=10, there are 4 achiral (AAAA, ABBA, BAAB, BBBB) and 6 chiral pairs (AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, BABB-BBAB). - Robert A. Russell, Nov 14 2018

For n > 0, a(2n+1) is the number of non-isomorphic 6C_m-snakes, where m = 2n+1 or m = 2n (for n>=2). A kC_n-snake is a connected graph in which the k>=2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - Christian Barrientos, May 15 2019

Number of achiral colorings of the edges of a tetrahedron with n available colors. - Robert A. Russell, Sep 07 2019

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of k-block partitions of an n-set up to reflection.

T(n,k) = pi_k(P_n) which is the number of non-equivalent partitions of the path on n vertices, with exactly k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Mohammad Hadi Shekarriz, Aug 21 2019

f and g are in the same class if function g(i) = f(n+1-i) for all i.

Decomposition by class size

.

n 1 2

---------------

1 1 0

2 2 1

3 9 9

4 16 120

5 125 1500

6 216 23220

7 2401 410571

.

Demonstration for the formula: the classes are either of size 1 or 2.

The classes of size 1 is for functions invariant by reversal. They are specified by half their values, including one more if n is odd. Their number is n^(ceiling(n/2)).

So the number of classes under this symmetry is half (the number of functions + the number of classes of size 1).

a(n) is the number of unoriented length n strings with a maximum of n colors. - Andrew Howroyd, Sep 13 2019

"BIK" (reversible, indistinct, unlabeled) transform of 3, 0, 0, 0, ...

a(n) is the dimension of the homogeneous component of degree n of the free unital special Jordan algebra on 3 generators (this follows from Cohn 1959). Note that this is no longer true for 4 generators and further. - Vladimir Dotsenko, Mar 31 2025

Number of periodic palindromes of length n using a maximum of k different symbols.

From Petros Hadjicostas, Sep 02 2018: (Start)

According to Christian Bower's theory of transforms, we have boxes of different sizes and different colors. The size of a box is determined by the number of balls it can hold. In this case, we assume all balls are the same and are unlabeled. Assume also that the number of possible colors a box with m balls can have is given by c(m). We place the boxes on a circle at equal distances from each other. Two configurations of boxes on the circle are considered equivalent if one can be obtained from the other by rotation. We are interested about circular configurations of boxes that are circular palindromes (i.e., necklaces with boxes that are the same when turned over). Let b(n) be the number of circularly palindromic configurations of boxes on a circle when the total number of balls in the boxes is n (and each box contains at least one ball).

Bower calls the sequence (b(n): n >= 1), the CPAL ("circular palindrome") transform of the input sequence (c(m): m >= 1). If the g.f. of the input sequence (c(m): m >= 1) is C(x) = Sum_{m>=1} c(m)*x^m, then the g.f. of the output sequence (b(n): n >= 1) is B(x) = Sum_{n >= 1} b(n)*x^n = (1 + C(x))^2/(2*(1 - C(x^2))) - 1/2.

In the current sequence, each box holds only one ball but can have one of k colors. Hence, c(1) = k but c(m) = 0 for m >= 2. Thus, C(x) = k*x. Then, for fixed k, the output sequence is (b(n): n >= 1) = (T(n, k): n >= 1), where T(n, k) = number of necklaces with n beads and k colors that are the same when turned over. If we let B_k(x) = Sum_{n>=1} T(n, k)*x^n, then B_k(x) = (1 + k*x)^2/(2*(1 - k*x^2)) - 1/2. From this, we can easily prove the formulae below.

Note that T(n, k=2) - 1 is the total number of Sommerville symmetric cyclic compositions of n. See pp. 301-304 in his paper in the links below. To see why this is the case, we use MacMahon's method of representing a cyclic composition of n with a necklace of 2 colors (see p. 273 in Sommerville's paper where the two "colors" are an x and a dot . rather than B and W). Given a Sommerville symmetrical composition b_1 + ... + b_r of n (with b_i >= 1 for all i and 1 <= r <= n), create the following circularly palindromic necklace with n beads of 2 colors: Start with a B bead somewhere on the circle and place b_1 - 1 W beads to the right of it; place a B bead to the right of the W beads (if any) followed by b_2 - 1 W beads; and so on. At the end, place a B bead followed with b_r - 1 W beads. (If b_i = 1 for some i, then a B bead follows a B bead since there are 0 W beads between them.) We thus get a circularly palindromic necklace with n beads of two colors. (The only necklace we cannot get with this method is the one than has all n beads colored W.)

It is interesting that the representation of a necklace of length n, say s_1, s_2, ..., s_n, as a periodic sequence (..., s_{-2}, s_{-1}, s_0, s_1, s_2, ...) with the property s_i = s_{i+n} for all i, as was done by Marks R. Nester in Chapter 2 of his 1999 PhD thesis, was considered by Sommerville in his 1909 paper (in the very first paragraph of his paper). (End)

Reversing the string does not leave it unchanged. Only one string from each pair is counted.

Equivalently, the number of nonequivalent strings up to reversal that are not palindromes.

Except for the first term, column k is the "BHK" (reversible, identity, unlabeled) transform of k,0,0,0,... [Corrected by Petros Hadjicostas, Jul 01 2018]

From Petros Hadjicostas, Jul 01 2018: (Start)

Consider the input sequence (c_k(n): n >= 1) with g.f. C_k(x) = Sum_{n>=1} c_k(n)*x^n. Let a_k(n) = BHK(c_k(n): n >= 1) be the output sequence under Bower's BHK transform. It can be proved that the g.f. of BHK(c_k(n): n >= 1) is A_k(x) = (C_k(x)^2 - C_k(x^2))/(2*(1-C_k(x))*(1-C_k(x^2))) + C_k(x). (See the comments for sequences A032096, A032097, and A032098.)

For column k of this two-dimensional array, the input sequence is defined by c_k(1) = k and c_k(n) = 0 for n >= 1. Thus, C_k(x) = k*x, and hence the g.f. of column k (with the term C_k(x) = k*x excluded) is (C_k(x)^2 - C_k(x^2))/(2*(1-C_k(x))*(1-C_k(x^2))) = (1/2)*(k - 1)*k*x^2/((k*x^2 - 1)*(k*x - 1)), from which we can easily prove Howroyd's formula.

(End)

Comment from Bahman Ahmadi, Aug 05 2019: (Start)

We give an alternative definition for the square array A(n,k) = T(n,k) with n >= 2 and k >= 0. A(n,k) is the number of inequivalent "distinguishing colorings" of the path on n vertices using at most k colors. The rows are indexed by n, the number of vertices of the path, and the columns are indexed by k, the number of permissible colors.

A vertex-coloring of a graph G is called "distinguishing" if it is only preserved by the identity automorphism of G. This notion is considered in the context of "symmetry breaking" of simple (finite or infinite) graphs. Two vertex-colorings of a graph are called "equivalent" if there is an automorphism of the graph which preserves the colors of the vertices. Given a graph G, we use the notation Phi_k(G) to denote the number of inequivalent distinguishing colorings of G with at most k colors. This sequence gives A(n,k) = Phi_k(P_n), i.e., the number of inequivalent distinguishing colorings of the path P_n on n vertices with at most k colors.

For n=3, we can color the vertices of P_3 with at most 2 colors in 3 ways such that all the colorings distinguish the graph (i.e., no non-identity automorphism of the path P_3 preserves the coloring) and that all the three colorings are inequivalent.

We have Phi_k(P_n) = binomial(k,2)*k^(n-2) + k*Phi_k(P_(n-2)) for n >= 4; Phi_k(P_2) = binomial(k,2); Phi_k(P_3) = k*binomial(k,2).

(End)

Also the number of 4-ary strings of length m = n+1 with number of 1's, 2's and 3's all even. Bijective proof, anyone? - Frank Ruskey, Jul 14 2002

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.