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A032190 Number of cyclic compositions of n into parts >= 2. 3
0, 1, 1, 2, 2, 4, 4, 7, 9, 14, 18, 30, 40, 63, 93, 142, 210, 328, 492, 765, 1169, 1810, 2786, 4340, 6712, 10461, 16273, 25414, 39650, 62074, 97108, 152287, 238837, 375166, 589526, 927554, 1459960, 2300347, 3626241, 5721044, 9030450, 14264308, 22542396 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Number of ways to partition n elements into pie slices each with at least 2 elements.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

C. G. Bower, Transforms (2)

P. Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 764

Index entries for sequences related to necklaces

FORMULA

"CIK" (necklace, indistinct, unlabeled) transform of 0, 1, 1, 1...

From Petros Hadjicostas, Sep 10 2017: (Start)

For all the formulas below, assume n >= 1. Here, phi(n) = A000010(n) is Euler's totient function.

a(n) = (1/n) * Sum_{d|n} b(d)*phi(n/d), where b(n) = A001610(n-1).

a(n) = (1/n) * Sum_{d|n} phi(n/d)*(Fibonacci(d-1) + Fibonacci(d+1) - 1) (because of the equation a(n) = A000358(n) - 1 stated in the CROSSREFS section below).

G.f.: -x/(1-x) + Sum_(k>=1} phi(k)/k * log(1/(1-B(x^k))) where B(x) = x*(1+x). (This is a modification of a formula due to Joerg Arndt.)

G.f.: Sum_{k>=1} phi(k)/k * log((1-x^k)/(1-B(x^k))), which agrees with the one in the Encyclopedia of Combinatorial Structures, #764, above. (We have Sum_{n>=1} (phi(n)/n)*log(1-x^n) = -x/(1-x), which follows from the Lambert series Sum_{n>=1} phi(n)*x^n/(1-x^n) = x/(1-x)^2.)

Sum_{d|n} a(d)*d = n*Sum_{d|n} b(d)/d, where b(n) = A001610(n-1).

(End)

MAPLE

# formula (5.3) of Daryl Deford for "Number of distinct Lucas tilings of a 1 X n

# bracelet up to symmetry" in "Enumerating distinct chessboard tilings"

A032190 := proc(n)

    local a, i, d ;

    a := 0 ;

    for i from 0 to ceil((n-1)/2) do

        for d in numtheory[divisors](i) do

            if modp(igcd(i, n-i), d) = 0 then

                a := a+(numtheory[phi](d)*binomial((n-i)/d, i/d))/(n-i) ;

            end if;

        end do:

    end do:

    a ;

end proc:

seq(A032190(n), n=1..60) ; # R. J. Mathar, Nov 27 2014

MATHEMATICA

nn=40; Apply[Plus, Table[CoefficientList[Series[CycleIndex[CyclicGroup[n], s]/.Table[s[i]->x^(2i)/(1-x^i), {i, 1, n}], {x, 0, nn}], x], {n, 1, nn/2}]] (* Geoffrey Critzer, Aug 10 2013 *)

A032190[n_] := Module[{a=0, i, d, j, dd}, For[i=1, i <= Ceiling[(n-1)/2], i++, For[dd = Divisors[i]; j=1, j <= Length[dd], j++, d=dd[[j]]; If[Mod[GCD[i, n-i], d] == 0, a = a + (EulerPhi[d]*Binomial[(n-i)/d, i/d])/(n-i)]]]; a]; Table[A032190[n], {n, 1, 60}] (* Jean-François Alcover, Nov 27 2014, after R. J. Mathar *)

CROSSREFS

a(n) = A000358(n) - 1. Cf. A008965.

Sequence in context: A034396 A253412 A291148 * A222737 A005852 A274625

Adjacent sequences:  A032187 A032188 A032189 * A032191 A032192 A032193

KEYWORD

nonn

AUTHOR

Christian G. Bower

EXTENSIONS

Better name from Geoffrey Critzer, Aug 10 2013

STATUS

approved

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Last modified September 20 06:59 EDT 2018. Contains 315226 sequences. (Running on oeis4.)