A032248 "DHK[ 7 ]" (bracelet, identity, unlabeled, 7 parts) transform of 1,1,1,1,...

4, 10, 28, 56, 113, 197, 340, 544, 856, 1284, 1896, 2709, 3816, 5247, 7128, 9504, 12540, 16302, 21001, 26728, 33748, 42185, 52364, 64448, 78832, 95725, 115600, 138720, 165648, 196707, 232560, 273600, 320601, 374034, 434796, 503448, 581020, 668173, 766084

Offset: 10

Author

Christian G. Bower

Keywords

• nonn
• easy

Comments

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 = 7 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 in the documentations of sequences and A008804 and A032246 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 (in the documentations of sequences and A008804 and A032246) 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)

Links

• Colin Barker, Table of n, a(n) for n = 10..1000
• C. G. Bower, Transforms (2)
• Petros Hadjicostas, The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry, 2019.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,1,0,-1,8,-6,-6,8,0,-3,1).

Crossrefs

Cf. A001399, A008804, A032246, A032247, A032250. Column k = 7 of A180472.

Programs

Mathematica

LinearRecurrence[{3,0,-8,6,6,-8,1,0,-1,8,-6,-6,8,0,-3,1},{4,10,28,56,113,197,340,544,856,1284,1896,2709,3816,5247,7128,9504},40] (* Harvey P. Dale, Jul 08 2024 *)

PARI

Vec(x^10*(4 - 2*x - 2*x^2 + 4*x^3 + x^4 - 2*x^5 + x^6) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^40)) \\ Colin Barker, Feb 25 2019

Formula

G.f.: x^7*(1/(14*(1 - x)^7) - 1/((2*(1 - x))*(1 - x^2)^3) + 3/(7*(1 - x^7))). - Petros Hadjicostas, Feb 24 2019

a(n) = 3*a(n-1) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 8*a(n-6) + a(n-7) - a(n-9) + 8*a(n-10) - 6*a(n-11) - 6*a(n-12) + 8*a(n-13) - 3*a(n-15) + a(n-16) for n>25. - Colin Barker, Feb 25 2019