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A181314 a(n) = ADPE(n) is the total number of aperiodic k-double-palindromes of n up to cyclic equivalence, where 1 <= k <= n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

FORMULA

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)

PROG

(PARI) a(n)={sumdiv(n, d, moebius(n/d)*((3 + d%2)*2^(d\2-1) - 1)) - 1} \\ Andrew Howroyd, Sep 28 2019

CROSSREFS

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

KEYWORD

nonn

AUTHOR

John P. McSorley, Oct 12 2010

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, Sep 27 2019

STATUS

approved

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Last modified May 19 23:40 EDT 2022. Contains 353847 sequences. (Running on oeis4.)