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.

1, 17, 127, 1871, 13969, 205793, 1536463, 22635359, 168996961, 2489683697

Offset

1,2

Comments

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.

The corresponding perfect square sum is given in A336189.

Integers m such that A185505(m) is a square. - Michel Marcus, Jul 11 2020

Links

Table of n, a(n) for n=1..10.

Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020.

Formula

Conjectures from Colin Barker, Jul 11 2020: (Start)

G.f.: x*(1 + x)*(1 + 16*x + x^2) / (1 - 110*x^2 + x^4).

a(n) = 110*a(n-2) - a(n-4) for n>4.

(End)

Example

Board is numbered as follows:
.
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
  .
a(1) = 1 is a term as 1 = 1^2 is a perfect square.
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.

Prog

(PARI) isok(m) = issquare((7*m^4 + 5*m^2)/12); \\ Michel Marcus, Jul 11 2020

Crossrefs

Cf. A336189, A185505, A000290, A000124, A000217.

Sequence in context: A066453 A298838 A114756 * A159563 A341397 A229516

Adjacent sequences: A336183 A336184 A336185 * A336187 A336188 A336189

Keyword

nonn,more

Author

Scott R. Shannon and Eric Angelini, Jul 11 2020

Extensions

a(10) from Michel Marcus, Jul 11 2020

Status

approved