A261131 Number of ways to write n as the sum of 3 positive palindromes.

0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 13, 15, 14, 15, 14, 14, 12, 12, 9, 9, 8, 7, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 8, 7, 7, 7, 7, 7

Offset

0,6

Comments

Conjecture: a(n) > 0 for n > 2, i.e., every number greater than 2 can be written as the sum of 3 positive palindromes.

Links

Giovanni Resta, Table of n, a(n) for n = 0..10000

Example

a(28) = 5 since 28 can be expressed in 5 ways as the sum of 3 positive palindromes, namely, 28 = 22+5+1 = 22+4+2 = 22+3+3 = 11+11+6 = 11+9+8.

Maple

p:= proc(n) option remember; local i, s; s:= ""||n;
      for i to iquo(length(s), 2) do if
        s[i]<>s[-i] then return false fi od; true
    end:
h:= proc(n) option remember; `if`(n<1, 0,
     `if`(p(n), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(3):
seq(a(n), n=0..120);  # Alois P. Heinz, Sep 19 2018

Mathematica

pal=Select[Range@ 1000, (d = IntegerDigits@ #; d == Reverse@ d)&]; a[n_] := Length@ IntegerPartitions[n, {3}, pal]; a /@ Range[0, 1000]
Table[Count[IntegerPartitions[n, {3}], _?(AllTrue[#, PalindromeQ]&)], {n, 0, 90}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 26 2021 *)

Crossrefs

Cf. A002113, A260254, A261132.

Column k=3 of A319453.

Sequence in context: A047503 A228803 A037014 * A225061 A133223 A065003

Adjacent sequences: A261128 A261129 A261130 * A261132 A261133 A261134

Keyword

nonn,base,look

Author

Giovanni Resta, Aug 10 2015

Status

approved