OFFSET
0,5
COMMENTS
The binary quintic has four invariants of degrees 4, 8, 12, 18. Those of degrees 4, 8, 12 are algebraically independent, the one of degree 18 squares to an expression in the others. [A. E. Brouwer]
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
A. E. Brouwer, Invariants of binary forms
Marko V. Jaric and Joseph L. Birman, Calculation of the Molien generating function for invariants of space groups, J. Math. Phys. 18 (1977), 1459-1465.
Kleinschmidt, Axel; Verschinin, Valentin Tetrahedral modular graph functions. J. High Energy Phys. 2017, No. 9, Paper No. 155, 38 p. (2017), eq. (3.38)
Atsuto Seko, Atsushi Togo, Isao Tanaka, Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential, arXiv:1901.02118 [physics.comp-ph], 2019.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1).
FORMULA
From Michael Somos, Oct 30 2011: (Start)
Euler transform of length 18 sequence [ 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1].
G.f.: (1 + x^9) / ((1 - x^2) * (1 - x^4) * (1 - x^6)).
a(-3 - n) = a(n).
G.f.: (1-x^3+x^6)/((1+x+x^2)*(1+x^2)*(1+x)^2*(1-x)^3). - R. J. Mathar, Dec 18 2014
EXAMPLE
G.f. = 1 + x^2 + 2*x^4 + 3*x^6 + 4*x^8 + x^9 + 5*x^10 + x^11 + 7*x^12 + 2*x^13 + 8*x^14 + ...
G.f. = 1 + q^4 + 2*q^8 + 3*q^12 + 4*q^16 + q^18 + 5*q^20 + q^22 + 7*q^24 + 2*q^26 + 8*q^28 + ...
MAPLE
seq(coeff(series((x^9+1)/((-x^2+1)*(-x^4+1)*(-x^6+1)), x, n+1), x, n), n = 0 .. 70); # modified by G. C. Greubel, Aug 03 2019
MATHEMATICA
CoefficientList[Series[(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)), {x, 0, 70}], x] (* T. D. Noe, Oct 30 2011 *)
LinearRecurrence[{0, 1, 1, 1, -1, -1, -1, 0, 1}, {1, 0, 1, 0, 2, 0, 3, 0, 4}, 70] (* Harvey P. Dale, Apr 09 2019 *)
PROG
(PARI) {a(n) = round( (if( n%2, n-9, n) \ 2 + 3)^2 / 12)} /* Michael Somos, Oct 30 2011 */
(PARI) {a(n) = if( n<-1, n = -3 - n); polcoeff( (1+x^9)/(1-x^2)/(1-x^4)/(1-x^6) + x * O(x^n), n)} /* Michael Somos, Oct 30 2011 */
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Aug 03 2019
(Sage) ((1+x^9)/((1-x^2)*(1-x^4)*(1-x^6))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(GAP) a:=[1, 0, 1, 0, 2, 0, 3, 0, 4];; for n in [10..70] do a[n]:=a[n-2]+a[n-3] + a[n-4]-a[n-5]-a[n-6]-a[n-7]+a[n-9]; od; a; # G. C. Greubel, Aug 03 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved