cp's OEIS Frontend

The upper principal and intermediate convergents to 2^(1/2), beginning with

2/1, 3/2, 10/7, 17/12, 58/41, form a strictly decreasing sequence;

essentially, numerators=A143609 and denominators=A084068.

This is a square array related to the square array of nonnegative integers, A001477. Each row k contains the positive argument of the largest triangular number equal to or less than 14*k in column 0 and a corresponding 2nd-order recursive sequence G(k) in the rest of the row. Each second-order recursive series term G(i) corresponds to a(k,i+1). If the product 14*k appears in row "r" of the square array A001477, then the product of adjacent terms G(i)*G(i+1), if greater than (r^2 + 3*r - 2)/2, is always in row "r" of square array A001477. If the product is less than (r^2 + 3*r -2)/2 then assuming the row can take negative indices, the product can still be said to lie in the same row r. For instance, 0, 1, 3, and 6 are each a triangular number and appear as the first 4 terms of row 0 of square array A001477. Note that in the next row and to the left of the 1, 3, and 6 are 2, 4 and 7 so going down a row and to the left in the square array increases the value by 1. Going down to the next row and to the left again would be 3, 5, and 8 so 3 which is 2 more than 1 would be in row 2 if that row were made to take the indices (2,-1).

A property of this table is that a(k+1,i)-a(k,i) directly depends on the value of a(k+1,0)-a(k,0) in the same manner regardless of the value of k. For example, a(k,2+n) - a(k,2+n) = A001652(n) for n=0,1,2,3,... whereever a(k+1,0) - a(k,0) = 1.

Also, a(k+1,2+n) - a(k,2+n) is divisible by A143608(n) for n>0 for all k.

This is a table related to the square array of the nonnegative integers (A001477). Each row k contains A003056(14*k) in column 0 and a corresponding 2nd order recursive sequence G(k) beginning at position a(k,1). That is each term G(i) is a(k,i+1). If A002262(14*n) is "r", the product of adjacent terms G(i)*G(i+1) if greater than (r^2 + 3*r - 2)/2, is always in row "r" of the square array A001477. If the product is less than (r^2 + 3*r -2)/2, then the product could still be said to lie in the same row r since the product is equal to the sum of a triangular number + r, which is a property of all numbers in row r of the square array A002262.

A property of this table is that a(k+1,i)-a(k,i) directly depends on the value of a(k+1,0)-a(k,0) in the same manner regardless of the value of k. For instance, wherever a(k+1,0)-a(k,0) = 0, a(k+1,i+1)-a(k,i+1) = A212329. Also, a(k+1,n+2)-a(k,n+2) is divisible by A143608(n).

It appears that for n>0, A143608(n) divides a(n).

The sequence a(n)/A143608(n) appears to generate A001541 interleaved with A001653. - R. J. Mathar, Jul 04 2012

It also appears that if p equals a prime of the form 8*r +/- 3 then a(p + 1) == 0 (mod p); and that if p is a prime in the form of 8*r +/- 1 then a(p - 1) == 0 (mod p), inherited from A143608.

This is a table related to A001477 interpreted as a square array of the onnegative integers (A001477). Each row k contains A003056(14*k) in column 0 and a corresponding 2nd order recursive sequence G(k) beginning at position a(k,1) such that G(i) = a(k,i+1). If the product 14*k appears in row "r" of the square array A001477, then the product of adjacent terms G(i)*G(i+1) if greater than (r^2 + 3*r - 2)/2, is always in row "r" of square array A001477.

A property of this table is that a(k+1,i)-a(k,i) directly depends on the value of a(k+1,0)-a(k,0) in the same manner regardless of the value of k. For instance, a(k+1,i+1)-a(k,i+1 = A210695(i) if a(k + 1,0) - a(k,0) = 1; while a(k+1,i+1)-a(k,i+1 = A001108(i) if a(k+1,0) - a(k,0) = 0.

A related property is that a(k+1,1+n) - a(k,1+n) is divisible by A143608(n) for all k.

The first column (k=1) holds the interleaved integer square roots of these two "Half-Square" expressions in ascending order: floor(m^2/2 + 1) for m=>0 and floor(m^2/2 - 1) for m=>1. The second column (k=2) holds the value of m that yields the corresponding integer square root.

The value of m for row n (at n mod 3 = 2) is the value of the square root for the next row (at n mod 3 = 0) which uses the other expression.

There are twice as many results for the expression floor(m^2/2 + 1) as for floor(m^2/2 - 1), interleaved consistently as two of every three results (as shown in the example below).

The first column, for n mod 3 = 1, produces A001541.

The first column, for n mod 3 = 2, produces A001653.

NOTE: Interleaving of the two sequences above is A079496.

The first column, for n mod 3 = 0, produces A002315 (NSW Numbers).

NOTE: Interleaving of A001541 and A002315 is A001333.

The second column, for n mod 3 = 1, produces A005319.

The second column, for n mod 3 = 2, produces A002315 (again).

NOTE: Interleaving of the two sequences above is A143608.

The second column, for n mod 3 = 0, produces A075870.

NOTE: Interleaving of A005319 and A075870 is A052542 = 2*A000129 (Pell)

The row sums at n mod 3 = 1 and n mod 3 = 0 are used in the recursion to produce values in subsequent rows of the array for both columns.

For rows at n mod 3 = 2, the ascending interleaved combination of A(n,1) and the row sum (of the same row) produces A000129 (Pell Numbers).

Row sums also hold all the integer square roots (as given in A001542) of the Half-Squares, (A007590), at n mod 3 = 2, and the corresponding values of m in the next row at n mod 3 = 0, corresponding to A001541.

The value of the floor of half the row sum, for n mod 3 =0 and n mod 3 = 1, produces A048739, giving the partial sums of A000129 (Pell Numbers), for the Pell Numbers produced through the prior row at n mod 3 = 2.

The value of half the row sum, for n mod 3 = 2, produces A001109 (without its initial 0). This subsequence is also produced from finding the integer square roots of A083374. The value of the indices of that sequence where these roots occur is given by A002315 (NSW Numbers).

The differences of two entries in row n equals the row sum for row n-3, consistently for all rows n > 3.

The ratio of the two entries in the same row converges to sqrt(2).

The ratio of two entries in the same column (either k=1 or k=2) converge as follows:

A(k,n)/A(k,n-1)--> sqrt(2) for n mod 3 = 0,

--> sqrt(2) + 1 for n mod 3 = 1,

--> sqrt(2)/2 + 1 for n mod 3 = 2.

A(k,n)/A(k,n-3)--> sqrt(8) + 3 for n mod 3 = 0, 1, or 2,

That last line means: A001541, A001653, A002315, A005319 and A075870 all have the convergence ratio of sqrt(8) + 3 for adjacent terms. In addition alternating Pell Numbers also converge to that ratio.