

A032248


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


5



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
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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)
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.
Sequence in context: A222963 A257028 A174938 * A092504 A038579 A133726
Adjacent sequences: A032245 A032246 A032247 * A032249 A032250 A032251


KEYWORD

nonn,easy,changed


AUTHOR

Christian G. Bower


STATUS

approved



