A032246 "DHK[ 5 ]" (bracelet, identity, unlabeled, 5 parts) transform of 1,1,1,1,...

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, 21050, 22880, 24840, 26910

Offset: 8

Author

Christian G. Bower

Keywords

• nonn
• easy

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

From Petros Hadjicostas, Feb 24 2019: (Start)

When k is odd >= 3, the DHK[k] transform of sequence c = (c(n): n >= 1), whose g.f. is C(x) = Sum_{n>=1} c(n)*x^n, has g.f. Sum_{n>=1} (DHK[k] c)n*x^n = (1/2)*Sum{d|k} mu(d)*((1/k)*C(x^d)^(k/d) - C(x^d)*C(x^(2*d))^((k/d) - 1)/2)).

For the current sequence we have k = 5 and c(n) = 1 for all n >= 1. Hence, C(x) = x/(1-x) and A(x) = Sum_{n>=1} a(n)*x^n = (x^k/2)*Sum_{d|k} mu(d)*((1/k)*(1-x^d)^(-k/d) - (1-x^d)^(-1)*(1-x^(2*d))^(-((k/d) - 1)/2)).

The latter g.f. agrees with Herbert Kociemba's formula found below only when k is an odd prime. The reason is that (DHK[k] c)_n, with c=(1,1,1,...), is the number of aperiodic bracelets without reflection symmetry with k black beads and n-k white beads, while Herbert Kociemba's formula counts all (periodic and aperiodic) bracelets without reflection symmetry with k black beads and n-k white beads. Hence, in the case k is an odd prime, the two formulas agree.

When k is even, the g.f. of the DHK[k] transform of sequence c = (c(n): n >= 1) is much more complicated.

Note that Herbert Kociemba's formula for counting all (periodic and aperiodic) bracelets with no reflection symmetry is still valid even when k is even; e.g., see sequence A008804 for the case k=4. For k = 4, all bracelets with 4 black beads and n-k = n-4 white beads that have no reflection symmetry are aperiodic, but this is not true anymore for k even >= 6.

(End)

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 8..1007
• C. G. Bower, Transforms (2)
• Petros Hadjicostas, The aperiodic version of Herbert Kociemba's formula for bracelets.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,3,-3,-1,4,-1,-2,1).

Crossrefs

Cf. A001399, A008804, A032248. Column k = 5 of A180472.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 2*x^8/((1-x)^2*(1-x^2)^2*(1-x^5)) )); // G. C. Greubel, Feb 25 2019

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 *)
Drop[CoefficientList[Series[2*x^8/((1-x)^2*(1-x^2)^2*(1-x^5)), {x,0,50}], x], 8] (* G. C. Greubel, Feb 25 2019 *)

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 */

PARI

Vec(2*x^8/((1-x)^2*(1-x^2)^2*(1-x^5)) + O(x^40)) \\ Colin Barker, Mar 13 2019

Sage

a=(2*x^8/((1-x)^2*(1-x^2)^2*(1-x^5))).series(x, 50).coefficients(x, sparse=False); a[8:] # G. C. Greubel, Feb 25 2019

Formula

G.f.: 2*x^8/((1-x)^2*(1-x^2)^2*(1-x^5)).

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 * ((1/k)*Sum_{n|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