cp's OEIS Frontend

3*a(n)^2 + 1 is a square. Moreover, 3*a(n)^2 + 1 = (2*a(n) - a(n-1))^2.

Consecutive terms give nonnegative solutions to x^2 - 4*x*y + y^2 = 1. - Max Alekseyev, Dec 12 2012

Values y solving the Pellian x^2 - 3*y^2 = 1; corresponding x values given by A001075(n). Moreover, we have a(n) = 2*a(n-1) + A001075(n-1). - Lekraj Beedassy, Jul 13 2006

Number of spanning trees in 2 X n grid: by examining what happens at the right-hand end we see that a(n) = 3*a(n-1) + 2*a(n-2) + 2*a(n-3) + ... + 2*a(1) + 1, where the final 1 corresponds to the tree ==...=| !. Solving this we get a(n) = 4*a(n-1) - a(n-2).

Complexity of 2 X n grid.

A016064 also describes triangles whose sides are consecutive integers and in which an inscribed circle has an integer radius. A001353 is exactly and precisely mapped to the integer radii of such inscribed circles, i.e., for each term of A016064, the corresponding term of A001353 gives the radius of the inscribed circle. - Harvey P. Dale, Dec 28 2000

n such that 3*n^2 = floor(sqrt(3)*n*ceiling(sqrt(3)*n)). - Benoit Cloitre, May 10 2003

For n>0, ratios a(n+1)/a(n) may be obtained as convergents of the continued fraction expansion of 2+sqrt(3): either as successive convergents of [4;-4] or as odd convergents of [3;1, 2]. - Lekraj Beedassy, Sep 19 2003

Ways of packing a 3 X (2*n-1) rectangle with dominoes, after attaching an extra square to the end of one of the sides of length 3. With reference to A001835, therefore: a(n) = a(n-1) + A001835(n-1) and A001835(n) = 3*A001835(n-1) + 2*a(n-1). - Joshua Zucker and the Castilleja School Math Club, Oct 28 2003

a(n+1) is a Chebyshev transform of 4^n, where the sequence with g.f. G(x) is sent to the sequence with g.f. (1/(1+x^2))G(x/(1+x^2)). - Paul Barry, Oct 25 2004

This sequence is prime-free, because a(2n) = a(n) * (a(n+1)-a(n-1)) and a(2n+1) = a(n+1)^2 - a(n)^2 = (a(n+1)+a(n)) * (a(n+1)-a(n)). - Jianing Song, Jul 06 2019

Numbers such that there is an m with t(n+m) = 3*t(m), where t(n) are the triangular numbers A000217. For instance, t(35) = 3*t(20) = 630, so 35 - 20 = 15 is in the sequence. - Floor van Lamoen, Oct 13 2005

a(n) = number of distinct matrix products in (A + B + C + D)^n where commutator [A,B] = 0 but neither A nor B commutes with C or D. - Paul D. Hanna and Max Alekseyev, Feb 01 2006

For n > 1, middle side (or long leg) of primitive Pythagorean triangles having an angle nearing Pi/3 with larger values of sides. [Complete triple (X, Y, Z), X < Y < Z, is given by X = A120892(n), Y = a(n), Z = A120893(n), with recurrence relations X(i+1) = 2*{X(i) - (-1)^i} + a(i); Z(i+1) = 2*{Z(i) + a(i)} - (-1)^i.] - Lekraj Beedassy, Jul 13 2006

From Dennis P. Walsh, Oct 04 2006: (Start)

Number of 2 X n simple rectangular mazes. A simple rectangular m X n maze is a graph G with vertex set {0, 1, ..., m} X {0, 1, ..., n} that satisfies the following two properties: (i) G consists of two orthogonal trees; (ii) one tree has a path that sequentially connects (0,0),(0,1), ..., (0,n), (1,n), ...,(m-1,n) and the other tree has a path that sequentially connects (1,0), (2,0), ..., (m,0), (m,1), ..., (m,n). For example, a(2) = 4 because there are four 2 X 2 simple rectangular mazes:

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(End)

[1, 4, 15, 56, 209, ...] is the Hankel transform of [1, 1, 5, 26, 139, 758, ...](see A005573). - Philippe Deléham, Apr 14 2007

The upper principal convergents to 3^(1/2), beginning with 2/1, 7/4, 26/15, 97/56, comprise a strictly decreasing sequence; numerators=A001075, denominators=A001353. - Clark Kimberling, Aug 27 2008

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

A001353 and A001835 = bisection of continued fraction [1, 2, 1, 2, 1, 2, ...], i.e., of [1, 3, 4, 11, 15, 41, ...].

For n>0, a(n) equals the determinant of an (n-1) X (n-1) tridiagonal matrix with ones in the super and subdiagonals and (4, 4, 4, ...) as the main diagonal. [Corrected by Johannes Boot, Sep 04 2011]

A001835 and A001353 = right and next to right borders of triangle A125077. (End)

a(n) is equal to the permanent of the (n-1) X (n-1) Hessenberg matrix with 4's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - John M. Campbell, Jun 09 2011

2a(n) is the number of n-color compositions of 2n consisting of only even parts; see Guo in references. - Brian Hopkins, Jul 19 2011

Pisano period lengths: 1, 2, 6, 4, 3, 6, 8, 4, 18, 6, 10, 12, 12, 8, 6, 8, 18, 18, 5, 12, ... - R. J. Mathar, Aug 10 2012

From Michel Lagneau, Jul 08 2014: (Start)

a(n) is defined also by the recurrence a(1)=1; for n>1, a(n+1) = 2*a(n) + sqrt(3*a(n)^2 + 1) where a(n) is an integer for every n. This sequence is generalizable by the sequence b(n,m) of parameter m with the initial condition b(1,m) = 1, and for n > 1 b(n+1,m) = m*b(n,m) + sqrt((m^2 - 1)*b(n,m)^2 + 1) for m = 2, 3, 4, ... where b(n,m) is an integer for every n.

The first corresponding sequences are

b(n,2) = a(n) = A001353(n);

b(n,3) = A001109(n);

b(n,4) = A001090(n);

b(n,5) = A004189(n);

b(n,6) = A004191(n);

b(n,7) = A007655(n);

b(n,8) = A077412(n);

b(n,9) = A049660(n);

b(n,10) = A075843(n);

b(n,11) = A077421(n);

....................

We obtain a general sequence of polynomials {b(n,x)} = {1, 2*x, 4*x^2 - 1, 8*x^3 - 4*x, 16*x^4 - 12*x^2 + 1, 32*x^5 - 32*x^3 + 6*x, ...} with x = m where each b(n,x) is a Gegenbauer polynomial defined by the recurrence b(n,x)- 2*x*b(n-1,x) + b(n-2,x) = 0, the same relation as the Chebyshev recurrence, but with the initial conditions b(x,0) = 1 and b(x,1) = 2*x instead b(x,0) = 1 and b(x,1) = x for the Chebyshev polynomials. (End)

If a(n) denotes the n-th term of the above sequence and we construct a triangle whose sides are a(n) - 1, a(n) + 1 and sqrt(3a(n)^2 + 1), then, for every n the measure of one of the angles of the triangle so constructed will always be 120 degrees. This result of ours was published in Mathematics Spectrum (2012/2013), Vol. 45, No. 3, pp. 126-128. - K. S. Bhanu and Dr. M. N. Deshpande, Professor (Retd), Department of Statistics, Institute of Science, Nagpur (India).

For n >= 1, a(n) equals the number of 01-avoiding words of length n - 1 on alphabet {0, 1, 2, 3}. - Milan Janjic, Jan 25 2015

For n > 0, 10*a(n) is the number of vertices and roots on level n of the {4, 5} mosaic (see L. Németh Table 1 p. 6). - Michel Marcus, Oct 30 2015

(2 + sqrt(3))^n = A001075(n) + a(n)*sqrt(3), n >= 0; integers in the quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 16 2018

A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Dec 12 2019

The Cholesky decomposition A = C C* for tridiagonal A with A[i,i] = 4 and A[i+1,i] = A[i,i+1] = -1, as it arises in the discretized 2D Laplace operator (Poisson equation...), has nonzero elements C[i,i] = sqrt(a(i+1)/a(i)) = -1/C[i+1,i], i = 1, 2, 3, ... - M. F. Hasler, Mar 12 2021

The triples (a(n-1), 2a(n), a(n+1)), n=2,3,..., are exactly the triples (a,b,c) of positive integers a < b < c in arithmetic progression such that a*b+1, b*c+1, and c*a+1 are perfect squares. - Bernd Mulansky, Jul 10 2021

From Greg Dresden and Linyun Sheng, Jul 01 2025: (Start)

a(n) is the number of ways to tile this strip of length n,

.________________

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||__||__||__|_\,

where the last cell is a right triangle, with three types of tiles: 1 X 1 squares, 1 X 1 small right triangles, and large right triangles (with large side length 2) formed by joining two of those small right triangles along a short leg. As an example, here is one of the a(7)=2911 ways to tile the 1 X 7 strip with these kinds of tiles:

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|\/_|\|/|__|/_\,

(End)

Kreweras calls this the complexity of the n X n grid.

a(n) is the number of perfect mazes made from a grid of n X n cells. - Leroy Quet, Sep 08 2007

Also number of domino tilings of the (2n-1) X (2n-1) square with upper left corner removed. For n=2 the 4 domino tilings of the 3 X 3 square with upper left corner removed are:

. ._. . ._. . ._. . ._.

.|__| .|__| .| | | .|___|

| |_| | | | | | ||| |_| |

||__| |||_| ||__| |_|_| - Alois P. Heinz, Apr 15 2011

Indeed, more is true. Let L denote the (2*n - 1) X (2*n - 1) square lattice graph with vertices (i,j), 1 <= i,j <= 2*n-1. Call a vertex (i,j) odd if both coordinates i and j are odd. Then there is a bijection between the set of spanning trees on the square n X n grid and the set of domino tilings of L with an odd boundary point removed. See Tzeng and Wu, 2002. This is a divisibility sequence, i.e., if n divides m then a(n) divides a(m). - Peter Bala, Apr 29 2014

Also, a(n) is the order of the sandpile group of the (n-1)X(n-1) grid graph. This is because the n X n grid is dual to (n-1)X(n-1) grid + sink vertex, and the latter is related to the sandpiles by the burning bijection. See Járai, Sec. 4.1, or Redig, Sec. 2.2. In M. F. Hasler's comment below, index n refers to the size of the grid underlying the sandpile. - Andrey Zabolotskiy, Mar 27 2018

From M. F. Hasler, Mar 07 2018: (Start)

The sandpile addition (+) of two n X n matrices is defined as the ordinary addition, followed by the topple-process in which each element larger than 3 is decreased by 4 and each of its von Neumann neighbors is increased by 1.

For any n, there is a neutral element e_n such that the set S(n) = { A in M_n({0..3}) | A (+) e_n = A } of matrices invariant under sandpile addition of e_n, forms a group, i.e., each element A in S(n) has an inverse A' in S(n) such that A (+) A' = e_n. (For n > 1, e_n cannot be the zero matrix O_n, because for this choice S(n) would include, e.g., the all 1's matrix 1_n which cannot have an inverse X such that 1_n (+) X = O_n. The element e_n is the unique nonzero matrix such that e_n (+) e_n = e_n.)

The present sequence lists the size of the abelian group (S(n), (+), e_n). See the example section for the e_n. The elements of S(2) are listed as A300006 and their inverses are listed as A300007. (End)

Also the number of connected induced (non-null) subgraphs of the grid graph P_m X P_n.

All rows (or columns) are linear recurrences with constant coefficients and the order of the recurrence of row m is at most 1 + A378941(m+1). At least for columns up to 7, this bound gives the actual order of the recurrence. The second differences of any column give those arrays that touch the top and bottom boundaries and have a recurrence order of 2 less since a finite state machine to enumerate these does not require states for empty rows. The number of states required is also considered in A140662 but does not take symmetry into account. - Andrew Howroyd, Dec 18 2024

a(n) is a divisibility sequence - m divides n implies that a(m) divides a(n). - Paul Raff, Mar 06 2009

Also number of domino tilings of the 5 X (2n-1) rectangle with upper left corner removed. For n=2 the 15 domino tilings of the 5 X 3 rectangle with upper left corner removed are:

. ._. . ._. . ._. . ._. . ._. . ._. . ._. . ._.

.|__| .| | | .|___| .|__| .|__| .| | | .| | | .|__|

| |_| | ||| | | | | | |_| | |_| | ||| | ||| | | | |

||__| ||__| |||_| || | | ||___| || | | ||___| |||_|

| |_| | |_| | |_| | ||| | | | | | ||| | | | | | | | |

||__| ||__| ||__| ||__| |||_| ||__| |||_| |||_|

. ._. . ._. . ._. . ._. . ._. . ._. . ._.

.|__| .|__| .|__| .|__| .|__| .|__| ._| | |

|_| | | | | | |_| | |_| | |_| | | |_| | |||

|_|_| |||_| | | || |__|_| |_|_| ||__| ||__|

|_| | |_| | ||| | | | | | | |_| |_| | |_| |

|_|_| |_|_| |_|_| |||_| ||__| |_|_| |_|_|

- Alois P. Heinz, Apr 14 2011

Also number of domino tilings of the 7 X (2n-1) rectangle with upper left corner removed. - Alois P. Heinz, Apr 14 2011

A linear divisibility sequence of order 8; a(n) divides a(m) whenever n divides m. It is the product of a 2nd-order Lucas sequence and a 4th-order linear divisibility sequence. - Peter Bala, Apr 27 2014

Also number of domino tilings of the 9 X (2n-1) rectangle with upper left corner removed. - Alois P. Heinz, Apr 14 2011

A linear divisibility sequence of order 16; a(n) divides a(m) whenever n divides m. It is the product of two 4th-order linear divisibility sequences A143699 and A241606. - Peter Bala, Apr 26 2014

Also number of domino tilings of the 11 X (2n-1) rectangle with upper left corner removed. - Alois P. Heinz, Apr 14 2011

A linear divisibility sequence of order 32; a(n) divides a(m) whenever n divides m. It is the product of four linear divisibility sequences - three Lucas sequences of order 2 and one linear divisibility sequence of order 4. - Peter Bala, Apr 27 2014

Also T(m,n) except when m = n = 0 is the number of connected edge covers in the m X n grid graph.