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

Author

Giovanni Resta, Aug 10 2015

Keywords

• nonn
• base
• look

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.

The conjecture is true (see links). - Giorgos Kalogeropoulos, May 10 2025

Examples

			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.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 0..10000
• Javier Cilleruelo, Florian Luca, Lewis Baxter, Every positive integer is a sum of three palindromes, arXiv:1602.06208, 2017.

Crossrefs

Cf. A002113, A260254, A261132.

Column k=3 of A319453.

Programs

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 *)