cp's OEIS Frontend

A261422 Number of ordered triples (u,v,w) of palindromes such that u+v+w=n.

%I A261422 #46 Aug 14 2020 13:28:31
%S A261422 1,3,6,10,15,21,28,36,45,55,63,72,79,84,87,88,87,84,79,72,66,55,51,45,
%T A261422 40,36,33,31,30,30,30,33,27,34,33,33,33,33,33,33,33,33,36,27,39,36,36,
%U A261422 36,36,36,36,36,36,39,27,45,39,39,39,39,39,39,39,39,42,27,52,42,42,42,42,42,42,42,42,45
%N A261422 Number of ordered triples (u,v,w) of palindromes such that u+v+w=n.
%C A261422 It is known that a(n)>0 for all n.
%H A261422 N. J. A. Sloane, <a href="/A261422/b261422.txt">Table of n, a(n) for n = 0..10000</a> (See also the extended file with 200000 terms below)
%H A261422 William D. Banks, <a href="https://arxiv.org/abs/1508.04721">Every natural number is the sum of forty-nine palindromes</a>, arXiv:1508.04721 [math.NT], 2015. [Establishes the much weaker result that the coefficients of P(x)^49 are positive (see Formula section below).]
%H A261422 Javier Cilleruelo and Florian Luca, <a href="https://arxiv.org/abs/1602.06208">Every positive integer is a sum of three palindromes</a>, arXiv: 1602.06208 [math.NT], 2016-2017.
%H A261422 Erich Friedman, <a href="https://erich-friedman.github.io/mathmagic/0699.html">Problem of the Month (June 1999)</a>
%H A261422 N. J. A. Sloane, <a href="/A261422/a261422.txt">Table of n, a(n) for n=0..200000</a>
%H A261422 N. J. A. Sloane, <a href="/A261422/a261422_1.txt">Maple program to produce 200000 terms</a>
%F A261422 G.f. = P(x)^3, where P(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^11 + x^22 + x^33 + x^44 + x^55 + x^66 + x^77 + x^88 + x^99 + x^101 + x^111 + ... = Sum_{palindromes p} x^p.
%e A261422 4 can be written as a sum of three palindromes in 15 ways: 4+0+0 (3 ways), 3+1+0 (6 ways), 2+2+0 (3 ways), and 2+1+1 (3 ways), so a(4)=15.
%t A261422 (* This program is not suitable to compute a large number of terms. *)
%t A261422 compositions[n_, k_] := Flatten[Permutations[PadLeft[#, k]]& /@ IntegerPartitions[n, k], 1];
%t A261422 a[n_] := Select[compositions[n, 3], AllTrue[#, PalindromeQ]&] // Length;
%t A261422 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Aug 05 2018 *)
%Y A261422 Cf. A002113. Differs from A261132, which assumes 0 <= u <= v <= w.
%Y A261422 For records see A262544, A262545.
%K A261422 nonn,look,hear,base
%O A261422 0,2
%A A261422 _N. J. A. Sloane_, Aug 27 2015