

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_{dk} 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/(1x) and A(x) = Sum_{n>=1} a(n)*x^n = (x^k/2)*Sum_{dk} mu(d)*((1/k)*(1x^d)^(k/d)  (1x^d)^(1)*(1x^(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 nk 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 nk 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 nk = n4 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(n1)  8*a(n3) + 6*a(n4) + 6*a(n5)  8*a(n6) + a(n7)  a(n9) + 8*a(n10)  6*a(n11)  6*a(n12) + 8*a(n13)  3*a(n15) + a(n16) 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



