A227972 Two column recursive array A(n,k), relating expressions based on half-squares (A007590) to each other and several other sequences, read by rows.

1, 0, 1, 1, 1, 2, 3, 4, 5, 7, 7, 10, 17, 24, 29, 41, 41, 58, 99, 140, 169, 239, 239, 338, 577, 816, 985, 1393, 1393, 1970, 3363, 4756, 5741, 8119, 8119, 11482, 19601, 27720, 33461, 47321, 47321, 66922, 114243, 161564, 195025, 275807, 275807, 390050, 665857, 941664, 1136689, 1607521

Offset: 1

Author

Richard R. Forberg, Aug 01 2013

Keywords

• nonn

Comments

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.

Examples

			The two column array with row number n and the row sum. An extra column on the right shows which expression is applicable to get that row's values: either floor(m^2/2 + 1) indicated as "+1",  or floor(m^2/2 - 1) indicated as "-1". (NOTE: The value of n is immaterial, except as a row number).
The array begins:
Row         k=1         k=2                   Applicable "Half-Square"
n          (sqrt)       (m)         Row Sum        Expression
1            1           0               1             +1
2            1           1               2             +1
3            1           2               3             -1
4            3           4               7             +1
5            5           7              12             +1
6            7          10              17             -1
7           17          24              41             +1
8           29          41              70             +1
9           41          58              99             -1
10          99         140             239             +1
11         169         239             408             +1
12         239         338             577             -1
13         577         816            1393             +1
14         985        1393            2378             +1
15        1393        1970            3363             -1
16        3363        4756            8119             +1
17        5741        8119           13860             +1
18        8119       11482           19601             -1
19       19601       27720           47321             +1
20       33461       47321           80782             +1
		

Crossrefs

Cf. A007590, A001541, A001653, A079496, A002315, A005319, A143608, A075870, A000129, A001109, A083374.

Formula

Initialize row 1 as A(1,1) = 1 and A(1,2) = 0, then:

For rows at n mod 3 = 0: A(n,1) = A(n-1, 2)

A(n,2) = A(n, 1) + A(n-2, 1)

For rows at n mod 3 = 1: A(n,1) = A(n-1, 1) + A(n-1, 2)

A(n,2) = A(n, 1) + A(n-1, 1)

For rows at n mod 3 = 2: A(n,1) = A(n-1,1) + A(n-3, 1)

A(n,2) = A(n-1,1) + A(n-1, 2)

Empirical g.f.: -x*(2*x^11-x^10-x^9+x^8-4*x^7+3*x^6-2*x^5-x^4-x^3-x^2-1) / ((x^6-2*x^3-1)*(x^6+2*x^3-1)). - Colin Barker, Aug 08 2013

Extensions

Some additional comments by Richard R. Forberg, Aug 12 2013