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 A181111 'ADP(n,k)' triangle read by rows. ADP(n,k) is the number of aperiodic k-double-palindromes of n. 5
 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, 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, 40, 60, 60, 40, 40, 10, 0, 0, 10, 8, 44, 40, 94, 60, 88, 32, 46, 10, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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. 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, 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. Let ADP(n,k) denote the number of aperiodic k-double-palindromes of n. This sequence is the 'ADP(n,k)' triangle read by rows. REFERENCES John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 FORMULA T(n,k) = A180279(n,k) - A179519(n,k). - Andrew Howroyd, Sep 27 2019 EXAMPLE The triangle begins: 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 12 16 6 0 0 8 6 24 24 18 24 8 0 0 8 8 28 20 44 24 28 8 0 ... For example, row 8 is: 0 6 6 14 12 16 6 0. We have ADP(8,3)=6 because there are 6 aperiodic 3-double-palindromes of 8: 116, 611, 224, 422, 233, and 332. 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. PROG (PARI) \\ here p(n, k) is A119963(n, k), q(n, k) is A051159(n-1, k-1). p(n, k) = { binomial((n-k%2)\2, k\2) } q(n, k) = { if(n%2==1&&k%2==0, 0, binomial((n-1)\2, (k-1)\2)) } 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 CROSSREFS Row sums are A181135. See sequence A180653 for the triangle whose (n, k) term gives the number of k-double-palindromes of n. See sequence A179519 for the triangle whose (n, k) term gives the number of aperiodic k-palindromes (single-palindromes) of n. Cf. A180279, A180918, A181169. Sequence in context: A333755 A238130 A238707 * A353856 A216800 A230831 Adjacent sequences: A181108 A181109 A181110 * A181112 A181113 A181114 KEYWORD nonn,tabl AUTHOR John P. McSorley, Oct 03 2010 EXTENSIONS a(37) corrected and terms a(56) and beyond from Andrew Howroyd, Sep 27 2019 STATUS approved

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Last modified December 3 18:40 EST 2023. Contains 367540 sequences. (Running on oeis4.)