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A181135
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a(n) = ADP(n) is the total number of aperiodic k-double-palindromes of n, where 2 <= k <= n.
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3
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0, 0, 2, 4, 12, 16, 42, 60, 112, 168, 310, 432, 756, 1106, 1722, 2640, 4080, 6062, 9198, 13860, 20300, 31062, 45034, 68340, 98208, 149940, 212576, 325080, 458724, 700128, 983010, 1501440, 2096096, 3202800, 4456074, 6800976, 9437148, 14398958, 19920474
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OFFSET
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1,3
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COMMENTS
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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, so it is an aperiodic 7-double-palindrome of 17. See sequence A181111.
The n-th term of this sequence is ADP(n), the total number of aperiodic k-double-palindromes of n, where 2 <= k <= n.
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:
(with k=2) 15=1|5, 51=5|1, 24=2|4, 42=4|2,
(with k=3) 114=11|4, 411=4|11,
(with k=4) 1113=111|3, 3111=3|111, 1311=131|1, 1131=1|131, 1122=11|22, and 2211=22|11,
(with k=5) 11112=1111|2, 21111=2|1111, 12111=121|11, 11121=11|121.
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REFERENCES
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John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010.
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LINKS
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FORMULA
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PROG
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(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
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CROSSREFS
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Row sums of A181111 (number of aperiodic k-double-palindromes of n).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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