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
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
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
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