A337842 The smallest palindrome with exactly n circular loops (or holes) in its decimal representation.

1, 0, 8, 606, 88, 808, 888, 68086, 8888, 88088, 88888, 6880886, 888888, 8880888, 8888888, 688808886, 88888888, 888808888, 888888888, 68888088886, 8888888888, 88888088888, 88888888888, 6888880888886, 888888888888, 8888880888888, 8888888888888, 688888808888886

Offset: 0

Author

Bernard Schott, Sep 25 2020

Keywords

• nonn
• base
• easy

Comments

The decimal digits 1, 2, 3, 5, 7 have no hole, and 4 is represented without a hole; otherwise, 0, 6, 9 have one hole each and 8 has two holes.

Least palindrome q such that A064532(q) = n.

Except for a(0) = 1, each term has only digits 0, 6 or 8 in its decimal expansion.

Examples

			a(3) = 606 because 6 and 0 have each one circular loop for a total of 3.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to holes in digits
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,111,0,0,0,-1110,0,0,0,1000).

Crossrefs

Cf. A000533, A002113, A002282, A064532, A332180.

Cf. A331898 (similar for primes).

Programs

Mathematica

a[0] = 1; a[n_] := Switch[Mod[n, 4], ?EvenQ, 8*(10^(n/2) - 1)/9, 1, 8*(10^((n + 1)/2) - 9*10^((n - 1)/4) - 1)/9, 3, 8*(10^((n + 3)/2) - 9*10^((n + 1)/4) - 1)/9 - 2*(10^((n + 1)/2) + 1)]; Array[a, 28, 0] (* _Amiram Eldar, Sep 26 2020 *)
LinearRecurrence[{0,0,0,111,0,0,0,-1110,0,0,0,1000},{1,0,8,606,88,808,888,68086,8888,88088,88888,6880886,888888},30] (* Harvey P. Dale, Mar 13 2022 *)

Formula

a(2m) = A002282(m) for m >= 1.

a(4m+1) = A332180(m) for m >= 1.

a(4m+3) = 6 * A000533(2m+2) + 10 * A332180(m) for m >= 0.

Extensions

More terms from Amiram Eldar, Sep 25 2020