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
Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020.
Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020. [Cached copy]
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
KEYWORD
nonn,more
AUTHOR
Scott R. Shannon and Eric Angelini, Jul 11 2020
EXTENSIONS
a(10) from Michel Marcus, Jul 11 2020
STATUS
approved