A048411 Squares whose consecutive digits differ by 1.

0, 1, 4, 9, 121, 676, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321

Offset: 1

Author

Patrick De Geest, Apr 15 1999

Keywords

• nonn
• base
• more

Comments

a(14), if it exists, is > 10^34. - Lars Blomberg, Nov 25 2016

Is it true that all terms are palindromes? - Chai Wah Wu, Apr 06 2018

a(14), if it exists, is > 10^52. - Michael S. Branicky, Apr 22 2025

Crossrefs

Cf. A002477, A048412.

Cf. A010052; intersection of A033075 and A000290.

Programs

Haskell

a048411 n = a048411_list !! (n-1)
a048411_list = filter ((== 1) . a010052) a033075_list
-- Reinhard Zumkeller, Feb 21 2012

Mathematica

Select[Range[0, 10^7]^2, Or[# == 0, IntegerLength@ # == 1, Union@ Abs@ Differences@ IntegerDigits@ # == {1}] &] (* Michael De Vlieger, Nov 25 2016 *)

Python

from sympy.ntheory.primetest import is_square
def gen(d, s=None):
    if d == 0: yield tuple(); return
    if s == None:
        yield from [(i, ) + g for i in range(1, 10) for g in gen(d-1, s=i)]
    else:
        if s > 0: yield from [(s-1, ) + g for g in gen(d-1, s=s-1)]
        if s < 9: yield from [(s+1, ) + g for g in gen(d-1, s=s+1)]
def afind(maxdigits):
    print(0, end=", ")
    for d in range(1, maxdigits+1):
        for g in gen(d, s=None):
            t = int("".join(map(str, g)))
            if is_square(t): print(t, end=", ")
afind(17) # Michael S. Branicky, Sep 26 2021

Formula

a(n) = A048412(n)^2.