A218035 -id:A218035 - cp's OEIS frontend

A307717 Number of palindromic squares, k^2, of length n such that k is also palindromic.

4, 0, 2, 0, 5, 0, 3, 0, 8, 0, 5, 0, 13, 0, 9, 0, 22, 0, 16, 0, 37, 0, 27, 0, 60, 0, 43, 0, 93, 0, 65, 0, 138, 0, 94, 0, 197, 0, 131, 0, 272, 0, 177, 0, 365, 0, 233, 0, 478, 0, 300, 0, 613, 0, 379, 0, 772, 0, 471, 0, 957, 0, 577, 0, 1170, 0, 698, 0, 1413, 0

Offset: 1

Robert Price, Apr 23 2019

			There are only two palindromic squares of length 3 whose root is also palindromic. 11^2=121 and 22^2=484. Thus, a(3)=2.

Patrick De Geest, Palindromic Squares in bases 2 to 17
M. Kauers and C. Koutschan, Guessing with Little Data, arXiv:2202.07966 [cs.SC], 2022.
Bertrand Teguia Tabuguia and Wolfram Koepf, FPS In Action: An Easy Way To Find Explicit Formulas For Interlaced Hypergeometric Sequences, arXiv:2207.01031 [cs.SC], 2022.
Bertrand Teguia Tabuguia, Hypergeometric-Type Sequences, arXiv:2401.00256 [cs.SC], 2023.
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).

Cf. A002778, A002779, A034307, A034822, A218035.

Table[Length[Select[Range[If[n == 1, 0, Ceiling[Sqrt[10^(n - 1)]]], Floor[Sqrt[10^n]]], # == IntegerReverse[#] && #^2 == IntegerReverse[#^2] &]], {n, 15}]

From Christoph Koutschan, Feb 19 2022: (Start)
a(2n-1) = A218035(n).
a(n) is given by a quasi-polynomial (for a proof, see A218035):
a(1) = 4;
a(2n) = 0;
a(4n+1) = (n^3-3*n^2+11*n+6)/3 (n > 0);
a(4n+3) = (n^3+5*n+12)/6 (n >= 0). (End)

a(16)-a(20) from Robert Price, Apr 25 2019

a(21)-a(70) from Giovanni Resta, Apr 28 2019

A343098 Number of palindromes < 10^n whose squares are also palindromes.

Original entry on oeis.org

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

Chai Wah Wu, Apr 04 2021

Partial sum of A218035. Number of terms in A057135 < 10^n.

			a(2) = 6 since the only palindromes < 100 whose square are palindromes are 0,1,2,3,11,22.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A000290, A002113, A002779, A057136, A057135, A218035.

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])

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

cp's OEIS Frontend

A307717 Number of palindromic squares, k^2, of length n such that k is also palindromic.

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