This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181135 #14 Oct 02 2019 15:00:24
%S A181135 0,0,2,4,12,16,42,60,112,168,310,432,756,1106,1722,2640,4080,6062,
%T A181135 9198,13860,20300,31062,45034,68340,98208,149940,212576,325080,458724,
%U A181135 700128,983010,1501440,2096096,3202800,4456074,6800976,9437148,14398958,19920474
%N A181135 a(n) = ADP(n) is the total number of aperiodic k-double-palindromes of n, where 2 <= k <= n.
%C A181135 A k-composition of n is an ordered collection of k positive integers (parts) which sum to n. A k-composition is aperiodic (primitive) if its period is k, i.e., if it is not the concatenation of at least two smaller compositions. A palindrome is a word which is the same when written backwards.
%C A181135 A k-double-palindrome of n is a k-composition of n which is the concatenation of two palindromes, PP'=P|P', where both |P|, |P'|>=1. For example 1123532=11|23532 is a 7-double-palindrome of 17 since both 11 and 23532 are palindromes. It is also aperiodic, so it is an aperiodic 7-double-palindrome of 17. See sequence A181111.
%C A181135 The n-th term of this sequence is ADP(n), the total number of aperiodic k-double-palindromes of n, where 2 <= k <= n.
%C A181135 For example ADP(6)=16 because there are 16 aperiodic k-double-palindromes of 6 for k=2,3,4,5, or 6. They are:
%C A181135 (with k=2) 15=1|5, 51=5|1, 24=2|4, 42=4|2,
%C A181135 (with k=3) 114=11|4, 411=4|11,
%C A181135 (with k=4) 1113=111|3, 3111=3|111, 1311=131|1, 1131=1|131, 1122=11|22, and 2211=22|11,
%C A181135 (with k=5) 11112=1111|2, 21111=2|1111, 12111=121|11, 11121=11|121.
%D A181135 John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010.
%H A181135 Andrew Howroyd, <a href="/A181135/b181135.txt">Table of n, a(n) for n = 1..1000</a>
%F A181135 a(n) = A180322(n) - A179781(n). - _Andrew Howroyd_, Sep 27 2019
%o A181135 (PARI) a(n) = { sumdiv(n, d, moebius(n/d) * (n * if(d%2, 1, 3/2) * 2^((d-1)\2) - 2^(d\2) )) } \\ _Andrew Howroyd_, Sep 27 2019
%Y A181135 Row sums of A181111 (number of aperiodic k-double-palindromes of n).
%Y A181135 Cf. A179781, A180322.
%K A181135 nonn
%O A181135 1,3
%A A181135 _John P. McSorley_, Oct 05 2010
%E A181135 a(8) corrected and a(11) and beyond from _Andrew Howroyd_, Sep 27 2019