Thats cool I get what you're saying there. I just checked and saw that sqrt(10368/2) = 72 also.
What do you mean by "triangular" number though. Like is it the c for a^2+b^2=c^2? I googled it and it doesn't match the definition google gave me.
Also noticed that if you (for c=145) take 72 and subtract your original n, you get your original x.
goal (1, 5, 12, 7, 5, 29)
start (1, 61, 12, 11, 1, 145)
sqare (0, 10368, 145, 144, 1, 21025)
n = 10368
sqrt(n/2) = 72.0
sqrt(n/2) - start[n] = 11.0
>The sequence of triangular numbers (sequence A000217 in the OEIS), starting at the 0th triangular number, is
>0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666…
I don't see 72 or 71 here so thats why I was confused.
Here we have
(e,n,d,d-1,1,c)
Then if we do our c^2 record we get
(0,(c^2+1)/2 - c, c, c-1, 1, c^2) N= (c^2+1)/2 - c
Then we can see that sqrt(N/2) - n = x
sqrt( ((c^2-2c+1)/2)/2 ) - n = x
sqrt( ((c-1)(c-1)/2)/2 ) - n = x
(c-1)/2 - n = x
c - 1 - 2n = 2x
c - 1 = 2x + 2n
c - 1 = 2(x+n)
c = 1 + 2x + 2n
c = a + 2x + 2n
nothing new