A008804 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^4)).

1, 2, 4, 6, 10, 14, 20, 26, 35, 44, 56, 68, 84, 100, 120, 140, 165, 190, 220, 250, 286, 322, 364, 406, 455, 504, 560, 616, 680, 744, 816, 888, 969, 1050, 1140, 1230, 1330, 1430, 1540, 1650, 1771, 1892, 2024, 2156, 2300, 2444, 2600, 2756, 2925, 3094, 3276, 3458

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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

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

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

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

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

Examples

			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 + ...
There are 10 asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to 7 under action of D_4:
[0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]
[1 6] [2 5] [3 4] [2 4] [3 3] [4 2] [5 1] [3 2] [4 1] [2 3]
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• C. G. Bower, Transforms (2)
• Petros Hadjicostas, The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry, 2019.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 197
• Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From _R. J. Mathar_, Nov 08 2008]
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Crossrefs

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

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

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)^2*(1-x^2)*(1-x^4)) )); // G. C. Greubel, Sep 12 2019

Maple

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

Mathematica

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

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

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 */

Sage

def A008804_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)^2*(1-x^2)*(1-x^4))).list()
A008804_list(60) # G. C. Greubel, Sep 12 2019

Formula

For a formula for a(n) see A014557.

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

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

G.f.: 1/((1-x)^4*(1+x)^2*(1+x^2)). - Jaume Oliver Lafont, Sep 20 2009

Euler transform of length 4 sequence [2, 1, 0, 1]. - Michael Somos, Feb 05 2011

a(n) = -a(-8 - n) for all n in Z. - Michael Somos, Feb 05 2011

From Herbert Kociemba, Nov 27 2016: (Start)

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.

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)

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

a(n) = A001400(n) + A001400(n-1) + A001400(n-2). - David García Herrero, Aug 26 2024

a(n) = floor((2*n^3 + 24*n^2 + n*(85+3*(-1)^n) + 96) / 96). - Hoang Xuan Thanh, May 24 2025