A378908 Square array, read by descending antidiagonals, where each row n comprises the integers w >= 1 such that A000037(n)*w^2+4 is a square.

4, 24, 2, 140, 8, 1, 816, 30, 3, 4, 4756, 112, 8, 40, 6, 27720, 418, 21, 396, 96, 2, 161564, 1560, 55, 3920, 1530, 12, 12, 941664, 5822, 144, 38804, 24384, 70, 456, 6, 5488420, 21728, 377, 384120, 388614, 408, 17316, 120, 1, 31988856, 81090, 987, 3802396

Offset: 1

Author

Charles L. Hohn, Dec 10 2024

Keywords

• nonn
• tabl

Comments

Also, integers w >= 1 for each row n >= 1 such that z+(1/z) is an integer, where x = A000037(n), y = w*sqrt(x), and z = (y+ceiling(y))/2.

All terms of row n are positive integer multiples of T(n, 1).

Limit_{k->oo} T(n, k+1)/T(n, k) = (sqrt(b^2-4)+b)/2 where b=T(n, 2)/T(n, 1).

Examples

			n=row index; x=nonsquare integer of index n (A000037(n)):
 n  x    T(n, k)
------+---------------------------------------------------------------------
 1  2 |  4,   24,   140,     816,      4756,       27720,        161564, ...
 2  3 |  2,    8,    30,     112,       418,        1560,          5822, ...
 3  5 |  1,    3,     8,      21,        55,         144,           377, ...
 4  6 |  4,   40,   396,    3920,     38804,      384120,       3802396, ...
 5  7 |  6,   96,  1530,   24384,    388614,     6193440,      98706426, ...
 6  8 |  2,   12,    70,     408,      2378,       13860,         80782, ...
 7 10 | 12,  456, 17316,  657552,  24969660,   948189528,   36006232404, ...
 8 11 |  6,  120,  2394,   47760,    952806,    19008360,     379214394, ...
 9 12 |  1,    4,    15,      56,       209,         780,          2911, ...
10 13 |  3,   33,   360,    3927,     42837,      467280,       5097243, ...
11 14 |  8,  240,  7192,  215520,   6458408,   193536720,    5799643192, ...
12 15 |  2,   16,   126,     992,      7810,       61488,        484094, ...
13 17 | 16, 1056, 69680, 4597824, 303386704, 20018924640, 1320945639536, ...
14 18 |  8,  272,  9240,  313888,  10662952,   362226480,   12305037368, ...
...
		

Crossrefs

Rows from n = 1 (nonzero terms): A005319, A052530, A001906, A122652, A001080*2, A001542, A239365, A075844, A001353, A075835, A068204*2, A001090*2, A121740*2, A202299*2.

Cf. A000037, A002420, A033999, A048942, A180495, A298675.

Programs

PARI

row(n)={my(v=List()); for(t=3, oo, if((t^2-4)%x>0 || !issquare((t^2-4)/x), next); listput(v, sqrtint((t^2-4)/x)); break); listput(v, v[1]*sqrtint(v[1]^2*x+4)); while(#v<10, listput(v, v[#v]*(v[2]/v[1])-v[#v-1])); Vec(v)}
for(n=1, 20, x=n+floor(1/2+sqrt(n)); print (n, " ", x, " ", row(n)))

Formula

For x = A000037(n) (nonsquare integer of index n):

If x is not the sum of 2 squares, then T(n, 1) = A048942(n); otherwise, T(n, 1) is a positive integer multiple of A048942(n).

For j in {-2, 1, 2, 4}, if x-j is a square (except 2-2=0^2 or 5-1=2^2), then T(n, 1) = (4/abs(j))*sqrt(x-j) and T(n, 2) = T(n, 1)^3/(4/abs(j)) + sign(j)*2*T(n, 1).

For j in {1, 4}, if x+j is a square, then T(n, 1) = 2/sqrt(4/j) and T(n, 2) = (4/j)*sqrt(x+j).

For k >= 2, T(n, k) = T(n, k-1)*sqrt(T(n, 1)^2*x+4) - [k>=3]*T(n, k-2).

T(n, 2) = Sum_{i=0..oo}(T(n, 1)^(2-2*i) * x^((1-2*i)/2) * A002420(i) * A033999(i)).

If T(n, 1) is even, then T(n, 2) = T(n, 1)*A180495(n); if T(n, 1) is odd and x is even, then T(n, 2) = T(n, 1)*sqrt(A180495(n)+2); if T(n, 1) and x are both odd, then T(n, 2) is a factor of T(n, 1)*A180495(n).

For k >= 3, T(n, k) = T(n, k-1)*(T(n, 2)/T(n, 1)) - T(n, k-2) = T(n, 1)*A298675(T(n, 2)/T(n, 1), k-1) + T(n, k-2) = sqrt((A298675(T(n, 2)/T(n, 1), k)^2-4)/x).