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A032246 "DHK[ 5 ]" (bracelet, identity, unlabeled, 5 parts) transform of 1,1,1,1... 0
2, 4, 10, 16, 28, 42, 64, 90, 126, 168, 224, 288, 370, 462, 576, 704, 858, 1030, 1232, 1456, 1716, 2002, 2330, 2688, 3094, 3536, 4032, 4570, 5168, 5814, 6528, 7296, 8140, 9044, 10032, 11088, 12236, 13460, 14784, 16192, 17710, 19320 (list; graph; refs; listen; history; text; internal format)
OFFSET

8,1

COMMENTS

a(n) is the number of bracelets with k=5 black beads and n-k white beads which have no reflection symmetry. - Herbert Kociemba, Nov 27 2016

LINKS

Table of n, a(n) for n=8..49.

C. G. Bower, Transforms (2)

FORMULA

G.f.: 2/((1-x^2)^2(1-x^5)(1-x)^2).

From Herbert Kociemba, Nov 27 2016: (Start)

More generally gf(k) is the g.f. for the number of bracelets without reflection symmetry with k black beads and n-k white beads.

gf(k): x^k/2 * ( Sum_{n, n divides k} phi(n)/(1-x^n)^(k/n) - (1+x)/(1-x^2)^floor(k/2+1) ). (End)

a(n) = a(5-n) for all n in Z. - Michael Somos, Nov 28 2016

0 = a(n) - 2*a(n+1) - a(n+2) + 4*a(n+3) - a(n+4) - 3*a(n+5) + 3*a(n+6) + a(n+7) - 4*a(n+8) + a(n+9) + 2*a(n+10) - a(n+11) for all n in Z. - Michael Somos, Nov 28 2016

EXAMPLE

G.f. = 2*x^8 + 4*x^9 + 10*x^10 + 16*x^11 + 28*x^12 + 42*x^13 + 64*x^14 + ...

MATHEMATICA

gf[x_, k_]:=x^k/2 (1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])-(1+x)/(1-x^2)^Floor[k/2+1]); CoefficientList[Series[gf[x, 5], {x, 0, 50}], x] (* Herbert Kociemba, Nov 27 2016 *)

PROG

(PARI) {a(n) = if( n<0, n=5-n); polcoeff( 2 * x^8 / ((1-x)^2*(1-x^2)^2*(1-x^5)) + x * O(x^n), n)}; /* Michael Somos, Nov 28 2016 */

CROSSREFS

Sequence in context: A144834 A006584 A280186 * A219901 A141138 A077627

Adjacent sequences:  A032243 A032244 A032245 * A032247 A032248 A032249

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified February 25 21:11 EST 2018. Contains 299657 sequences. (Running on oeis4.)