This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378908 #16 Jan 03 2025 02:18:32
%S A378908 4,24,2,140,8,1,816,30,3,4,4756,112,8,40,6,27720,418,21,396,96,2,
%T A378908 161564,1560,55,3920,1530,12,12,941664,5822,144,38804,24384,70,456,6,
%U A378908 5488420,21728,377,384120,388614,408,17316,120,1,31988856,81090,987,3802396
%N 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.
%C A378908 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.
%C A378908 All terms of row n are positive integer multiples of T(n, 1).
%C A378908 Limit_{k->oo} T(n, k+1)/T(n, k) = (sqrt(b^2-4)+b)/2 where b=T(n, 2)/T(n, 1).
%F A378908 For x = A000037(n) (nonsquare integer of index n):
%F A378908 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).
%F A378908 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).
%F A378908 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).
%F A378908 For k >= 2, T(n, k) = T(n, k-1)*sqrt(T(n, 1)^2*x+4) - [k>=3]*T(n, k-2).
%F A378908 T(n, 2) = Sum_{i=0..oo}(T(n, 1)^(2-2*i) * x^((1-2*i)/2) * A002420(i) * A033999(i)).
%F A378908 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).
%F A378908 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).
%e A378908 n=row index; x=nonsquare integer of index n (A000037(n)):
%e A378908 n x T(n, k)
%e A378908 ------+---------------------------------------------------------------------
%e A378908 1 2 | 4, 24, 140, 816, 4756, 27720, 161564, ...
%e A378908 2 3 | 2, 8, 30, 112, 418, 1560, 5822, ...
%e A378908 3 5 | 1, 3, 8, 21, 55, 144, 377, ...
%e A378908 4 6 | 4, 40, 396, 3920, 38804, 384120, 3802396, ...
%e A378908 5 7 | 6, 96, 1530, 24384, 388614, 6193440, 98706426, ...
%e A378908 6 8 | 2, 12, 70, 408, 2378, 13860, 80782, ...
%e A378908 7 10 | 12, 456, 17316, 657552, 24969660, 948189528, 36006232404, ...
%e A378908 8 11 | 6, 120, 2394, 47760, 952806, 19008360, 379214394, ...
%e A378908 9 12 | 1, 4, 15, 56, 209, 780, 2911, ...
%e A378908 10 13 | 3, 33, 360, 3927, 42837, 467280, 5097243, ...
%e A378908 11 14 | 8, 240, 7192, 215520, 6458408, 193536720, 5799643192, ...
%e A378908 12 15 | 2, 16, 126, 992, 7810, 61488, 484094, ...
%e A378908 13 17 | 16, 1056, 69680, 4597824, 303386704, 20018924640, 1320945639536, ...
%e A378908 14 18 | 8, 272, 9240, 313888, 10662952, 362226480, 12305037368, ...
%e A378908 ...
%o A378908 (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)}
%o A378908 for(n=1, 20, x=n+floor(1/2+sqrt(n)); print (n, " ", x, " ", row(n)))
%Y A378908 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.
%Y A378908 Cf. A000037, A002420, A033999, A048942, A180495, A298675.
%K A378908 nonn,tabl
%O A378908 1,1
%A A378908 _Charles L. Hohn_, Dec 10 2024