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A343098
Number of palindromes < 10^n whose squares are also palindromes.
1
1, 4, 6, 11, 14, 22, 27, 40, 49, 71, 87, 124, 151, 211, 254, 347, 412, 550, 644, 841, 972, 1244, 1421, 1786, 2019, 2497, 2797, 3410, 3789, 4561, 5032, 5989, 6566, 7736, 8434, 9847, 10682, 12370, 13359, 15356, 16517, 18859, 20211, 22936, 24499, 27647, 29442, 33055
OFFSET
0,2
COMMENTS
Partial sum of A218035. Number of terms in A057135 < 10^n.
FORMULA
a(n) = #{i:A057135(i)<10^n}.
For n > 0, a(n) = Sum_{i=1..n} A218035(i).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: (-x^9 + x^7 - x^6 - 6*x^5 - x^4 + 7*x^3 + 2*x^2 - 3*x - 1)/((x - 1)^5*(x + 1)^4).
a(n) = 1491 + 904*n + 510*n^2 - 52*n^3 + 6*n^4 + (-1)^n * (45 - 296*n + 42*n^2 - 4*n^3) for n>0. - Greg Dresden, Jun 20 2021
EXAMPLE
a(2) = 6 since the only palindromes < 100 whose square are palindromes are 0,1,2,3,11,22.
PROG
(Python)
A343098_list = [1, 4, 6, 11, 14, 22, 27, 40, 49, 71]
for i in range(10**3): A343098_list.append(A343098_list[-1] + 4*A343098_list[-2] - 4*A343098_list[-3] - 6*A343098_list[-4] + 6*A343098_list[-5] + 4*A343098_list[-6] - 4*A343098_list[-7] - A343098_list[-8] + A343098_list[-9])
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Chai Wah Wu, Apr 04 2021
STATUS
approved