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 A180249 a(n) is the total number of k-reverses of n. 4
 1, 2, 4, 8, 16, 26, 50, 80, 130, 212, 342, 518, 820, 1276, 1864, 2960, 4336, 6704, 9710, 15068, 21368, 33420, 47082, 72950, 102316, 158888, 220882, 342616, 475108, 734816, 1015778, 1569680, 2161944, 3337952, 4587200, 7069748, 9699292, 14932444, 20445520 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS See sequence A180171 for the definition of a k-reverse of n. Briefly, a k-reverse of n is a k-composition of n whose reverse is cyclically equivalent to itself. This sequence is the total number of k-reverses of n for k=1,2,...,n. It is the row sums of the 'R(n,k)' triangle from sequence A180171. For example a(6)=26 because there are 26 k-reverses of n=6 for k=1,2,3,4,5, or 6. They are, in cyclically equivalent, classes: {6}, {15,51}, {24,42},{33},{114,411,141},{222} {1113,3111,1311,1131}, {1122,2112,2211,1221}, {1212,2121}, {11112,21111,12111,11211,11121}, {111111}. REFERENCES John P. McSorley: Counting k-compositions with palindromic and related structures. Preprint, 2010. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 FORMULA a(n) = Sum_{d|n} d*A056493(d)/2. - Andrew Howroyd, Oct 07 2017 From Petros Hadjicostas, Oct 15 2017: (Start) a(n) = (n/2)*Sum_{d|n} (phi^(-1)(d)/d)*b(n/d), where phi^(-1)(n) = A023900(n) is the Dirichlet inverse of the Euler totient function and b(n) = A029744(n+1) (= 3*2^((n/2)-1), if n is even, and = 2^((n+1)/2), if n is odd). G.f.: Sum_{n>=1} phi^(-1)(n)*g(x^n), where phi^(-1)(n) = A023900(n) and g(x) = x*(x+1)*(2*x+1)/(1-2*x^2)^2. (End) MATHEMATICA f[n_Integer] := Block[{c = 0, k = 1, ip = IntegerPartitions@ n, lmt = 1 + PartitionsP@ n, ipk}, While[k < lmt, c += g[ ip[[k]]]; k++ ]; c]; g[lst_List] := Block[{c = 0, len = Length@ lst, per = Permutations@ lst}, While[ Length@ per > 0, rl = Union[ RotateLeft[ per[], # ] & /@ Range@ len]; If[ MemberQ[rl, Reverse@ per[]], c += Length@ rl]; per = Complement[ per, rl]]; c]; Array[f, 24] (* Robert G. Wilson v, Aug 25 2010 *) b[n_] := Sum[MoebiusMu[n/d] * If[OddQ[d], 2, 3] * 2^Quotient[d-1, 2], {d, Divisors[n]}]; a[n_] := Sum[d*b[d], {d, Divisors[n]}] / 2; Array[a, 39] (* Jean-François Alcover, Nov 04 2017, after Andrew Howroyd *) PROG (PARI) \\ here b(n) is A056493 b(n) = sumdiv(n, d, moebius(n/d) * if(d%2, 2, 3) * 2^((d-1)\2)); a(n) = sumdiv(n, d, d*b(d)) / 2; \\ Andrew Howroyd, Oct 07 2017 CROSSREFS If we ask for the number of cyclically equivalent classes we get sequence A052955. For example the 6th term of A052955 is 11, corresponding to the 11 classes in the example above. Row sums of A180171. Cf. A056493, A180322. Sequence in context: A070789 A302588 A319385 * A060957 A322326 A018826 Adjacent sequences:  A180246 A180247 A180248 * A180250 A180251 A180252 KEYWORD nonn AUTHOR John P. McSorley, Aug 19 2010 EXTENSIONS a(11) - a(24) from Robert G. Wilson v, Aug 25 2010 a(25) - a(27) from Robert G. Wilson v, Aug 29 2010 Terms a(28) and beyond from Andrew Howroyd, Oct 07 2017 STATUS approved

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Last modified May 12 19:15 EDT 2021. Contains 343829 sequences. (Running on oeis4.)