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A180750
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a(n) = DP(n) is the total number of k-double-palindromes of n, where 2 <= k <= n.
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1
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0, 1, 3, 6, 13, 21, 43, 68, 116, 185, 311, 464, 757, 1157, 1741, 2720, 4081, 6214, 9199, 14078, 20353, 31405, 45035, 68930, 98224, 150761, 212706, 326362, 458725, 702209, 983011, 1504400, 2096441, 3207137, 4456139, 6808172, 9437149, 14408669, 19921297, 30393800
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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 palindrome is a word which is the same when written backwards. A k-double-palindrome of n (see sequence A180653) 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.
The n-th term of this sequence is DP(n), the total number of k-double-palindromes of n, where 2 <= k <= n.
For example, DP(6)=21 because there are 21 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, 33=3|3,
(with k=3) 114=11|4, 411=4|11, 222=2|22,
(with k=4) 1113=111|3, 3111=3|111, 1311=131|1, 1131=1|131, and 1122=11|22, 2211=22|11, 1212=121|2, 2121=2|121,
(with k=5) 11112=1111|2, 21111=2|1111, 12111=121|11, 11121=11|121,
(with k=6) 111111=1|11111.
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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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G.f.: Sum_{n>=1} phi^{(-1)}(n)*f(x^n) - Sum_{n>=1} mu(n)*g(x^n), where phi^{(-1)}(n) = A023900(n) is the Dirichlet inverse of Euler's totient function, mu(n) = A008683(n) is the Mobius function, f(x) = x*(x+1)*(2*x+1)/(1-2*x^2)^2, and g(x) = x*(1+2*x)/(1-2*x^2). - Petros Hadjicostas, Nov 06 2017
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CROSSREFS
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a(n) is the sum of the n-th row of the triangle sequence A180653 (number of k-double-palindromes of n).
The n-th term of sequence A016116 is the total number of k-palindromes (single palindromes) of n.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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