A219032 Number of distinct squares as subwords of decimal representation of n-th square.

1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 3, 2, 3, 4, 3, 2, 2, 2, 2, 3, 3, 3, 2, 2, 1, 2, 1, 2, 2, 3, 3, 3, 4, 4, 2, 4, 3, 4, 4, 2, 4, 4, 4, 5, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 5, 4, 4, 3, 2, 2, 2, 4, 4, 2, 3, 2, 3, 6, 4, 3, 2, 2, 3, 1, 2, 3, 3, 5, 2, 2, 2, 2, 3

Offset: 0

Author

Reinhard Zumkeller, Nov 10 2012

Keywords

• nonn
• base

Comments

a(n) is the number of squares in n-th row of triangle A219031.

Examples

			.   n   row n in A219031
.  -----------------------------
.  20   [0, 4, 40, 400]            a(20) = #{0, 4, 400} = 3;
.  21   [1, 4, 41, 44, 441]        a(21) = #{1, 4, 441} = 3;
.  22   [4, 8, 48, 84, 484]        a(22) = #{4, 484} = 2;
.  23   [2, 5, 9, 29, 52, 529]     a(23) = #{9, 529} = 2;
.  24   [5, 6, 7, 57, 76, 576]     a(24) = #{576} = 1;
.  25   [2, 5, 6, 25, 62, 625]     a(25) = #{25, 625} = 2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A010052, A120004, A219031.

Programs

Haskell

a219032 = sum . map a010052 . a219031_row

Python

from sympy import integer_nthroot
def A219032(n):
    s = str(n*n)
    m = len(s)
    return len(set(filter(lambda x: integer_nthroot(x,2)[1], (int(s[i:j]) for i in range(m) for j in range(i+1,m+1))))) # Chai Wah Wu, Oct 19 2021

Formula

a(n) = Sum_{k=0..A120004(n^2)} A010052(A219031(n,k)).