A298481 a(n) is the number of ways to partition the binary representation of n into the minimal number of palindromic parts.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1

Offset

1,10

Comments

A minimal palindromic partition is a partition of the string into palindromes with the fewest parts.

A298475(n) gives the size of the minimal partition of the binary representation of n.

Records occur at 1, 10, 42, 170, 682, 1357, 5428, 5453, 21812, 21837, 45746, ....

Links

Peter Kagey, Table of n, a(n) for n = 1..10000

Example

The a(n) minimal palindromic partitions for five integers:
  n | a(n) | A298475(n) | binary   | partitions
----+------+------------+----------+--------------------------------------
2   | 1    | 2          | 10       | 1'0
5   | 1    | 1          | 101      | 101
10  | 2    | 2          | 1010     | 101'0 or 1'010
37  | 2    | 3          | 100101   | 1001'0'1 or 1'00'101
149 | 3    | 3          | 10010101 | 1001'010'1, 1'00'10101, or 1001'0'101

Mathematica

{1, 1}~Join~Array[Function[w, Length@ MinimalBy[#, Length] &@ Select[#, And[AllTrue[#, PalindromeQ], Union@ Map[Length, #] != {1}] &] &@ Union@ Map[Select[SplitBy[#, IntegerQ], IntegerQ@ First@ # &] &, Map[Insert[w, ".", #] &, Map[{#} &, Rest@ Subsets@ Range@ Length@ w, {2}]]]]@ IntegerDigits[#, 2] &, 103, 3] (* Michael De Vlieger, Jan 23 2018 *)

Crossrefs

Cf. A298475.

Sequence in context: A288120 A156264 A249770 * A324872 A307608 A323022

Adjacent sequences: A298478 A298479 A298480 * A298482 A298483 A298484

Keyword

nonn,base

Author

Peter Kagey, Jan 19 2018

Status

approved