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 A181111 #20 Oct 30 2021 07:17:39
%S A181111 0,0,0,0,2,0,0,2,2,0,0,4,4,4,0,0,4,2,6,4,0,0,6,6,12,12,6,0,0,6,6,14,
%T A181111 12,16,6,0,0,8,6,24,24,18,24,8,0,0,8,8,28,20,44,24,28,8,0,0,10,10,40,
%U A181111 40,60,60,40,40,10,0,0,10,8,44,40,94,60,88,32,46,10,0
%N A181111 'ADP(n,k)' triangle read by rows. ADP(n,k) is the number of aperiodic k-double-palindromes of n.
%C A181111 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.
%C A181111 A palindrome is a word which is the same when written backwards.
%C A181111 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.
%C A181111 For example 1123532=11|23532 is a 7-double-palindrome of 17 since both 11 and 23532 are palindromes. It is also aperiodic, and so it is an aperiodic 7-double-palindrome of 17. The 4-double-palindrome of 8 1313=131|3 is not aperiodic, so it is not an aperiodic 4-double-palindrome of 8.
%C A181111 Let ADP(n,k) denote the number of aperiodic k-double-palindromes of n.
%C A181111 This sequence is the 'ADP(n,k)' triangle read by rows.
%D A181111 John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010.
%H A181111 Andrew Howroyd, <a href="/A181111/b181111.txt">Table of n, a(n) for n = 1..1275</a>
%F A181111 T(n,k) = A180279(n,k) - A179519(n,k). - _Andrew Howroyd_, Sep 27 2019
%e A181111 The triangle begins:
%e A181111 0
%e A181111 0 0
%e A181111 0 2 0
%e A181111 0 2 2 0
%e A181111 0 4 4 4 0
%e A181111 0 4 2 6 4 0
%e A181111 0 6 6 12 12 6 0
%e A181111 0 6 6 14 12 16 6 0
%e A181111 0 8 6 24 24 18 24 8 0
%e A181111 0 8 8 28 20 44 24 28 8 0
%e A181111 ...
%e A181111 For example, row 8 is: 0 6 6 14 12 16 6 0.
%e A181111 We have ADP(8,3)=6 because there are 6 aperiodic 3-double-palindromes of 8: 116, 611, 224, 422, 233, and 332.
%e A181111 We have ADP(8,4)=14 because there are 14 4-double-palindromes of 8: 1115, 5111, 1511, 1151, 1214, 4121, 1412, 2141, 1133, 3311, 1232, 2123, 3212, and 2321.
%o A181111 (PARI) \\ here p(n,k) is A119963(n,k), q(n,k) is A051159(n-1, k-1).
%o A181111 p(n, k) = { binomial((n-k%2)\2, k\2) }
%o A181111 q(n, k) = { if(n%2==1&&k%2==0, 0, binomial((n-1)\2, (k-1)\2)) }
%o A181111 T(n, k) = sumdiv(gcd(n, k), d, moebius(d) * (k*p(n/d, k/d) - q(n/d, k/d))); \\ _Andrew Howroyd_, Sep 27 2019
%Y A181111 Row sums are A181135.
%Y A181111 See sequence A180653 for the triangle whose (n, k) term gives the number of k-double-palindromes of n.
%Y A181111 See sequence A179519 for the triangle whose (n, k) term gives the number of aperiodic k-palindromes (single-palindromes) of n.
%Y A181111 Cf. A180279, A180918, A181169.
%K A181111 nonn,tabl
%O A181111 1,5
%A A181111 _John P. McSorley_, Oct 03 2010
%E A181111 a(37) corrected and terms a(56) and beyond from _Andrew Howroyd_, Sep 27 2019