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

0,3

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.

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).

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.

Example

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

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 *)

Crossrefs

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

Cf. A331898 (similar for primes).

Sequence in context: A269508 A090923 A336941 * A266317 A374142 A080320

Adjacent sequences: A337839 A337840 A337841 * A337843 A337844 A337845

Keyword

nonn,base,easy

Author

Bernard Schott, Sep 25 2020

Extensions

More terms from Amiram Eldar, Sep 25 2020

Status

approved