Javascript 66 bytes
Returns an array of strings representing the numbers in the sequence (necessary because 12345678987654321 is too large to be accurately represented as an int in Javascript)
o=>(n=0,Array(9).fill('').map(x=>(n=n*10+1,(x+n*n).replace(0,1))))
Explanation
o=>( // o is unused input (saves 1 byte vs ())
n=0, // n^2 is the rhombus number at each stage
Array(9) // create an array of length 9
.fill('') // fill it with empty strings
// we must fill it with something, because map skips
// empty entries
.map( // loop over the array and return a new array with the
// values returned by the callback
x=>( // x is the element in the array
n=n*10+1, // make n 1, 11, 111, 1111, etc
(x+n*n) // get the string value of n^2
.replace(0,1) // replace 0's with 1's (necessary for
// 12345678987654321, because
// 111111111^2 = 12345678987654320 in javascript)
)
)
)
