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 A336186 #38 Aug 20 2025 17:18:30
%S A336186 1,17,127,1871,13969,205793,1536463,22635359,168996961,2489683697
%N A336186 Side length of a square block of integers, with 1 at the top-left corner, on a diagonally numbered 2D board such that the sum of the integers in the square is a perfect square.
%C A336186 Consider a diagonally numbered 2D board shown in the example below. Draw a square, including the 1 at the top-left corner, around a block of integers and sum the integers within the square. This sequence gives the number of integers on the side of that square such that the resulting sum of integers is a perfect square.
%C A336186 The corresponding perfect square sum is given in A336189.
%C A336186 Integers m such that A185505(m) is a square. - _Michel Marcus_, Jul 11 2020
%H A336186 Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2020/06/prime-squares.html">Prime squares and square squares</a>, personal blog "Cinquante signes", Jun. 29, 2020.
%H A336186 Eric Angelini, <a href="/A336186/a336186.pdf">Prime squares and square squares</a>, personal blog "Cinquante signes", Jun. 29, 2020. [Cached copy]
%F A336186 Conjectures from _Colin Barker_, Jul 11 2020: (Start)
%F A336186 G.f.: x*(1 + x)*(1 + 16*x + x^2) / (1 - 110*x^2 + x^4).
%F A336186 a(n) = 110*a(n-2) - a(n-4) for n>4. (End)
%F A336186 Empirical from _Bill McEachen_, Aug 08 2025: (Start)
%F A336186 a(n) = floor(k1*B^(n+1)) for odd n, and floor(k2*B^n) for even n, where k1 =(26*sqrt(21)-119)/14, k2 = (2*sqrt(21)-7)/14, and B = sqrt(55 + 12*sqrt(21)).
%F A336186 Above closed-forms via Amiram Eldar equate to Barker's recurrence. (End)
%e A336186 Board is numbered as follows:
%e A336186 .
%e A336186 1 2 4 7 11 16 .
%e A336186 3 5 8 12 17 .
%e A336186 6 9 13 18 .
%e A336186 10 14 19 .
%e A336186 15 20 .
%e A336186 21 .
%e A336186 .
%e A336186 a(1) = 1 is a term as 1 = 1^2 is a perfect square.
%e A336186 a(2) = 17 is a term as the block of integers, with the seventeen numbers {1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137} along the top edge and the seventeen numbers {1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153} along the left edge, sum to 48841 = 221^2 which is a perfect square.
%o A336186 (PARI) isok(m) = issquare((7*m^4 + 5*m^2)/12); \\ _Michel Marcus_, Jul 11 2020
%Y A336186 Cf. A336189, A185505, A000290, A000124, A000217.
%K A336186 nonn,more
%O A336186 1,2
%A A336186 _Scott R. Shannon_ and _Eric Angelini_, Jul 11 2020
%E A336186 a(10) from _Michel Marcus_, Jul 11 2020