A392226 Squares that are the concatenation of x, 1 and x for some x.

69169, 76176, 12722025112722025, 23671716123671716, 26222400126222400, 41494116141494116, 50566689150566689, 59910025159910025, 71095881171095881, 99716676199716676, 111913916111119139161, 310524204913105242049, 371509802513715098025, 10920555895396110920555895396, 12194596259856112194596259856

Offset

1,1

Comments

Squares of the form (10^(d+1)+1)*x + 10^d where 10^(d-1) <= x < 10^d.

Links

Table of n, a(n) for n=1..15.

Example

a(3) = 12722025112722025 is a term because it is 112791955^2 and is the concatenation of 12722025, 1 and 12722025.

Maple

Res:= NULL: count:= 0:
for d from 2 to 14 do
  S:= [isolve(10^d + x*(10^(d+1)+1) = y^2)];
  S:= map(t -> subs(subs(_Z1=0, t), x), S);
  S:= sort(select(t -> t >= 10^(d-1) and t < 10^d, S));
  for s in S do
    t:= (10^(d+1)+1)*s+10^d; Res:= Res, t; count:= count+1;
   od
od:
Res;

Prog

(Python)
from math import isqrt
from itertools import count, islice
def b(n): return 10**len(str(n))*(10*n+1) + n  # A392225
def agen(): # generator of terms
    yield from (filter(lambda x: isqrt(x)**2 == x, (b(n) for n in count(1) if n%10 in {0, 1, 4, 5, 6, 9})))
g = agen()
[print(next(g), end=", ") for n in count(1)] # Michael S. Branicky, Jan 04 2026

Crossrefs

Intersection of A000290 and A392225.

Sequence in context: A096551 A096552 A385127 * A031663 A105010 A190379

Adjacent sequences: A392223 A392224 A392225 * A392227 A392228 A392229

Keyword

nonn,base

Author

Robert Israel, Jan 03 2026

Status

approved