cp's OEIS Frontend

Lim_{n-> infinity} a(n)/a(n+1) = 0.045361... = 1/(11+sqrt(122)) = (sqrt(122)-11).

Lim_{n-> infinity} a(n+1)/a(n) = 22.045361... = (11+sqrt(122)) = 1/(sqrt(122)-11).

Lim_{n -> infinity} a(n)/a(n+1) = 0.04339638... = 2/(23+sqrt(533)) = (sqrt(533)-23)/2.

Lim_{n -> infinity} a(n+1)/a(n) = 23.04339638... = (23+sqrt(533))/2 = 2/(sqrt(533) - 23).

Lim_{n->infinity} a(n)/a(n+1) = 0.0415945... = 1/(12+sqrt(145)) = sqrt(145) - 12.

Lim_{n->infinity} a(n+1)/a(n) = 24.0415945... = 12+sqrt(145) = 1/(sqrt(145)-12).

The left column, A(n,0), is A000129(n), Pell Numbers.

The top row, A(0,k), is A077957(k) plus an initial 0, which is the inverse binomial transform of A000129.

These may be considered initializing values, or results, depending the perspective taken, since there are several ways to generate the array. See Formula section for details.

The columns of the array hold all values, in sequential order, of numbers m such that 2m^2 + 2^k or 2m^2 - 2^k are squares, and their corresponding square roots in the next column, which then form the "next round" of m values for k+1.

For example A(n,0) are numbers such that 2m^2 +- 1 are squares, the integer square roots of each are in A(n,1), which are then numbers m such that 2m^2 +- 2 are squares, with those square roots in A(n,2), etc.

A(n, k)/A(n,k-2) = 2; A(n,k)/A(n,k-1) converges to sqrt(2) for large n.

A(n,k)/A(n-1,k) converges to 1 + sqrt(2) for large n.

The other columns of this array hold current OEIS sequences as follows:

A(n,1) = A001333(n); A(n,2) = A163271(n); A(n,3) = A002203(n);

Bisections of these column-oriented sequences also appear in the OEIS, corresponding to the even and odd rows of the array, which in turn correspond to the two different recursive square root equations in the formula section below.

Farey fraction denominators interleave columns 0 and 1, and the corresponding numerators interleave columns 1 and 2, for approximating sqrt(2). See A002965 and A119016, respectively.

The other rows of this array hold current OEIS sequences as follows:

A(1,k) = A016116(k); A(2,k) = A029744(k) less the initial 1;

A(3,k) = A070875(k); A(4,k) = A091523(k) less the initial 8.

The Pell Numbers (A000219) are the only initializing set of numbers where the two recursive square root equations (see below) produce exclusively integer values, for all iterations of k. For any other initial values only even iterations (at k = 2, 4, ...) produce integers.

The numbers in this array, especially the first three columns, are also integer square roots of these expressions: floor(m^2/2), floor(m^2/2 + 1), floor (m^2/2 - 1). See A227972 for specific arrangements and relationships. Also: ceiling(m^2/2), ceiling(m^2/2 + 1), ceiling (m^2/2 -1), m^2+1, m^2-1, m^2*(m^2-1)/2, m^2*(m^2-1)/2, in various different arrangements. Many of these involve: A000129(2n)/2 = A001109(n).

A001109 also appears when multiplying adjacent columns: A(n,k) * A(n,k+1) = (k+1) * A001109(n), for all k.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

Numbers m such that r = 24*m+1 and 2*r-1 are both squares. - Bruno Berselli, Jul 17 2014

From Johannes W. Meijer, Aug 01 2010: (Start)

The a(n) represent the number of n-move paths of a chess king on a 3 X 3 board that end or start in a given corner square m (m = 1, 3, 7, 9). To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the king's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.

Inverse binomial transform of A079291 (without the leading 0).

(End)

From R. J. Mathar, Oct 12 2010: (Start)

The row n=3 of an array counting king walks on an n X n board with k steps, starting from a corner:

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, ...;

1, 3, 18, 80, 400, 1904, 9248, 44544, 215296, 1039104, 5018112, ...;

1, 3, 18, 105, 615, 3600, 21075, 123375, 722250, 4228125, 24751875, ...;

1, 3, 18, 105, 684, 4359, 28278, 182349, 1179792, 7622667, 49283802, ...;

1, 3, 18, 105, 684, 4550, 30807, 209867, 1434279, 9815190, 67209723, ...;

1, 3, 18, 105, 684, 4550, 31340, 218056, 1533712, 10829360, 76720288, ...;

1, 3, 18, 105, 684, 4550, 31340, 219555, 1559835, 11177190, 80573373, ...;

1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11259785, 81765550, ...;

1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82025163, ...;

1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, ...;

1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, ...;

The partial sums along the rows are documented in A123109 (king walks with between 1 and k steps). (End)

Lim_{n-> infinity} a(n+1)/a(n) = 199.00502499874... = (199 + sqrt(39605))/2.

Lim_{n->infinity} a(n)/a(n+1) = 0.00502499874... = 2/(199 + sqrt(39605)) = (sqrt(39605) - 199)/2.

A floretion-generated sequence relating NSW and Pell numbers.

Elements of odd index in the sequence gives A002315. a(n+2) - a(n) = A002203(n+2).

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[B*C} with B = - .25'i + .25'j + .5'k - .25i' + .25j' + .5k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and C = + .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e