%I #125 Sep 03 2024 01:29:54

%S 1,2,4,6,10,14,20,26,35,44,56,68,84,100,120,140,165,190,220,250,286,

%T 322,364,406,455,504,560,616,680,744,816,888,969,1050,1140,1230,1330,

%U 1430,1540,1650,1771,1892,2024,2156,2300,2444,2600,2756,2925,3094,3276,3458

%N Expansion of 1/((1-x)^2*(1-x^2)*(1-x^4)).

%C b(n)=a(n-3) is the number of asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to n, under action of dihedral group D_4(b(0)=b(1)=b(2)=0). G.f. for b(n) is x^3/((1-x)^2*(1-x^2)*(1-x^4)). - _Vladeta Jovovic_, May 07 2000

%C If the offset is changed to 5, this is the 2nd Witt transform of A004526 [Moree]. - _R. J. Mathar_, Nov 08 2008

%C a(n) is the number of partitions of 2*n into powers of 2 less than or equal to 2^3. First differs from A000123 at n=8. - _Alois P. Heinz_, Apr 02 2012

%C a(n) is the number of bracelets with 4 black beads and n+3 white beads which have no reflection symmetry. For n=1 we have for example 2 such bracelets with 4 black beads and 4 white beads: BBBWBWWW and BBWBWBWW. - _Herbert Kociemba_, Nov 27 2016

%C a(n) is the also number of aperiodic bracelets with 4 black beads and n+3 white beads which have no reflection symmetry. This is equivalent to saying that a(n) is the (n+7)th element of the DHK[4] (bracelet, identity, unlabeled, 4 parts) transform of 1, 1, 1, ... (see Bower's link about transforms). Thus, for n >= 1 , a(n) = (DHK[4] c)_{n+7}, where c = (1 : n >= 1). This is because every bracelet with 4 black beads and n+3 white beads which has no reflection symmetry must also be aperiodic. This statement is not true anymore if we have k black beads where k is even >= 6. - _Petros Hadjicostas_, Feb 24 2019

%H T. D. Noe, <a href="/A008804/b008804.txt">Table of n, a(n) for n = 0..1000</a>

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%H Petros Hadjicostas, <a href="/A008804/a008804.pdf">The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry</a>, 2019.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=197">Encyclopedia of Combinatorial Structures 197</a>

%H Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From _R. J. Mathar_, Nov 08 2008]

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,2,-2,0,2,-1).

%F For a formula for a(n) see A014557.

%F a(n) = (84 +85*n +24*n^2 +2*n^3 +12*A056594(n+3) +3*(-1)^n*(n+4))/96. - _R. J. Mathar_, Nov 08 2008

%F a(n) = 2*(Sum_{k=0..floor(n/2)} A002620(k+2)) - A002620(n/2+2)*(1+(-1)^n)/2. - _Paul Barry_, Mar 05 2009

%F G.f.: 1/((1-x)^4*(1+x)^2*(1+x^2)). - _Jaume Oliver Lafont_, Sep 20 2009

%F Euler transform of length 4 sequence [2, 1, 0, 1]. - _Michael Somos_, Feb 05 2011

%F a(n) = -a(-8 - n) for all n in Z. - _Michael Somos_, Feb 05 2011

%F From _Herbert Kociemba_, Nov 27 2016: (Start)

%F 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.

%F 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) ). The g.f. here is gf(4)/x^7 because of the different offset. (End)

%F E.g.f.: ((48 + 54*x + 15*x^2 + x^3)*cosh(x) + 6*sin(x) + (36 + 57*x + 15*x^2 + x^3)*sinh(x))/48. - _Stefano Spezia_, May 15 2023

%F a(n) = A001400(n) + A001400(n-1) + A001400(n-2). - _David García Herrero_, Aug 26 2024

%e G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 14*x^5 + 20*x^6 + 26*x^7 + 35*x^8 + ...

%e There are 10 asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to 7 under action of D_4:

%e [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]

%e [1 6] [2 5] [3 4] [2 4] [3 3] [4 2] [5 1] [3 2] [4 1] [2 3]

%p seq(coeff(series(1/((1-x)^2*(1-x^2)*(1-x^4)), x, n+1), x, n), n = 0..60); # _G. C. Greubel_, Sep 12 2019

%t LinearRecurrence[{2,0,-2,2,-2,0,2,-1}, {1,2,4,6,10,14,20,26}, 60] (* _Vladimir Joseph Stephan Orlovsky_, Feb 23 2012 *)

%t 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,4]/x^7,{x,0,60}],x] (* _Herbert Kociemba_, Nov 27 2016 *)

%t Table[(84 +12*(-1)^n +85*n +3*(-1)^n*n +24*n^2 +2*n^3 +12*Sin[n Pi/2])/96, {n,0,60}] (* _Eric W. Weisstein_, Oct 12 2017 *)

%t CoefficientList[Series[1/((1-x)^4*(1+x)^2*(1+x^2)), {x,0,60}], x] (* _Eric W. Weisstein_, Oct 12 2017 *)

%o (PARI) a(n)=(84+12*(-1)^n+6*I*((-I)^n-I^n)+(85+3*(-1)^n)*n+24*n^2 +2*n^3)/96 \\ _Jaume Oliver Lafont_, Sep 20 2009

%o (PARI) {a(n) = my(s = 1); if( n<-7, n = -8 - n; s = -1); if( n<0, 0, s * polcoeff( 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; /* _Michael Somos_, Feb 02 2011 */

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)^2*(1-x^2)*(1-x^4)) )); // _G. C. Greubel_, Sep 12 2019

%o (Sage)

%o def A008804_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P(1/((1-x)^2*(1-x^2)*(1-x^4))).list()

%o A008804_list(60) # _G. C. Greubel_, Sep 12 2019

%o (GAP) a:=[1,2,4,6,10,14,20,26];; for n in [9..60] do a[n]:=2*a[n-1] -2*a[n-3]+2*a[n-4]-2*a[n-5]+2*a[n-7]-a[n-8]; od; a; # _G. C. Greubel_, Sep 12 2019

%Y Cf. A000123, A001400, A002620, A004526, A005232, A014557, A032246, A032248, A053307.

%Y Column k=3 of A181322. Column k = 4 of A180472 (but with different offset).

%K nonn,nice,easy

%O 0,2

%A _N. J. A. Sloane_