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 rows (see example) gives A225826.

This triangle is to A225826 as Losanitsch's triangle A034851 is to A005418.

By columns:

T(n,1) is A004526.

T(n,2) is A000217.

T(n,3) is A225972.

T(n,4) is A071239.

T(n,5) is A222715.

T(n,6) is A228581.

T(n,7) is A228582.

T(n,8) is A228583.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014

Sum of rows (see example) gives A225827.

This triangle is to A225827 as Losanitsch's triangle A034851 is to A005418, and triangle A226048 to A225826.

By columns:

T(n,1) is A052928.

T(n,2) is A226292.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 3 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

The length of row n is 4*n+1.

Sum of rows (see example) gives A225828.

This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, and triangle A226290 to A225827.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 4 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

The length of row n is 5*n+1.

Sum of rows (see example) gives A225829.

This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, and triangle A225812 to A225828.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 5 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

The length of row n is 6*n+1.

Sum of rows (see example) gives A225830.

This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, and triangle A228022 to A225829.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 6 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

The length of row n is 7*n+1.

Sum of rows (see example) gives A225831.

This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, triangle A228022 to A225829, and triangle A228165 to A225830.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 7 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

The length of row n is 8*n+1.

Sum of rows (see example) gives A225832.

This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, triangle A228022 to A225829, triangle A228165 to A225830, and triangle A228166 to A225831.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 8 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 28 2014

The length of row n is 9*n+1.

Sum of rows (see example) gives A225832.

This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, triangle A228022 to A225829, triangle A228165 to A225830, triangle A228166 to A225831, and triangle A228167 to A225832.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 9 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 27 2015

In the square table A000012, A005418, and A225826 to A225834 are the first 11 rows (see example).

In the square table, m odd (see formula). The order of the recurrence equations is 4. Let it be (a1(m),a2(m),a3(m),a4(m)) the characterizing 4-plet of a(m). The sequence a1(m) belongs to A028403 (2^m+2^((m+1)/2)), -a2(m) to A147538 (2^m*(2^((m+1)/2)-1)) and a4(m) to A013824 (2^(2m)*2^((m+1)/2)). -a3(m) sequence formula is 2^m*(2^m+2^((m+1)/2)).

All the coefficients of x in generating functions from A225826 to A225834 belong to A113979.