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A032248 "DHK[ 7 ]" (bracelet, identity, unlabeled, 7 parts) transform of 1,1,1,1,... 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

10,1

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).

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

PROG

(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

CROSSREFS

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

Sequence in context: A333172 A257028 A174938 * A092504 A038579 A133726

Adjacent sequences:  A032245 A032246 A032247 * A032249 A032250 A032251

KEYWORD

nonn,easy

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified July 14 07:02 EDT 2020. Contains 335718 sequences. (Running on oeis4.)