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A181314
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a(n) = ADPE(n) is the total number of aperiodic k-double-palindromes of n up to cyclic equivalence, where 1 <= k <= n.
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2
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0, 0, 1, 2, 5, 6, 13, 17, 27, 38, 61, 80, 125, 174, 245, 359, 509, 727, 1021, 1484, 2029, 3006, 4093, 6029, 8183, 12158, 16351, 24380, 32765, 48848, 65533, 97919, 131005, 196094, 262121, 392363, 524285, 785406, 1048445, 1571309, 2097149, 3143496, 4194301, 6288380, 8388323
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OFFSET
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1,4
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COMMENTS
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a(n) = ADPE(n) is the total number of aperiodic k-double-palindromes of n up to cyclic equivalence. See sequence A181169 for the definitions of an aperiodic k-double-palindrome of n and of cyclic equivalence.
Sequence A181169 is the 'ADPE(n,k)' triangle read by rows where ADPE(n,k) is the number of aperiodic k-double-palindromes of n up to cyclic equivalence.
For example, we have a(6) = ADPE(6) = ADPE(6,1) + ADPE(6,2) + ADPE(6,3) + ADPE(6,4) + ADPE(6,5) + ADPE(6,6) = 0 + 2 + 1 + 2 + 1 + 0 = 6. The 6 aperiodic double-palindromes of 6 up to cyclic equivalence are: 15, 24, 114, 1113, 1122, 11112. They are the representatives of the cyclic equivalence classes: {15,51}, {24,42}, {114,411,141},{1113,3111,1311,1131}, {1122,2112,2211,1221} and {11112,21111,12111,11211,11121}.
Hence a(n) = ADPE(n) is the total number of cyclic equivalence classes of compositions of n containing at least one aperiodic double-palindrome of n.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..1000
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FORMULA
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From Andrew Howroyd, Sep 28 2019: (Start)
a(n) = A056493(n) - 1 for n > 1.
G.f.: (x^2-2*x)/(1-x) + Sum_{k=1..n} mu(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)).
(End)
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PROG
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(PARI) a(n)={sumdiv(n, d, moebius(n/d)*((3 + d%2)*2^(d\2-1) - 1)) - 1} \\ Andrew Howroyd, Sep 28 2019
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CROSSREFS
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Row sums of A181169.
If we remove the aperiodic requirement, we get sequence A027383, see the comment from McSorley there. Also see sequences A181111 and A181135.
Cf. A056493.
Sequence in context: A283684 A325285 A323348 * A027010 A038191 A321472
Adjacent sequences: A181311 A181312 A181313 * A181315 A181316 A181317
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KEYWORD
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nonn
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AUTHOR
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John P. McSorley, Oct 12 2010
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EXTENSIONS
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Terms a(11) and beyond from Andrew Howroyd, Sep 27 2019
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STATUS
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approved
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