OFFSET
0,3
COMMENTS
This is the same as A008763 but without the four leading zeros. There are so many situations where one wants this sequence rather than A008763 that it seems appropriate for it to have its own entry.
But see A008763 (still the main entry) for numerous applications and references.
Also, Molien series for invariants of finite Coxeter group D_4 (bisected).
The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017
a(n) is the number of partitions of n into parts 1, 2, and 3, where there are two sorts of parts 2. - Joerg Arndt, Jun 27 2017
REFERENCES
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).
FORMULA
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8) for n>7. - Vincenzo Librandi, Jan 11 2016
a(n) = -a(-8-n) for all n in Z. - Michael Somos, Jun 26 2017
MATHEMATICA
CoefficientList[Series[1/((1-x)*(1-x^2)^2*(1-x^3)), {x, 0, 50}], x] (* JungHwan Min, Jan 10 2016 *)
LinearRecurrence[{1, 2, -1, -2, -1, 2, 1, -1}, {1, 1, 3, 4, 7, 9, 14, 17}, 100] (* Vincenzo Librandi, Jan 11 2016 *)
PROG
(PARI) Vec(1/((1-x)*(1-x^2)^2*(1-x^3)) + O(x^100)) \\ Michel Marcus, Jan 11 2016
(Magma) I:=[1, 1, 3, 4, 7, 9, 14, 17]; [n le 8 select I[n] else Self(n-1)+2*Self(n-2)-Self(n-3)-2*Self(n-4)-Self(n-5)+2*Self(n-6)+Self(n-7)-Self(n-8): n in [1..60]]; // Vincenzo Librandi, Jan 11 2016
(PARI) {a(n) = (9*(n+4)*(-1)^n + 2*n^3 + 24*n^2 + 87*n + 157) \ 144}; /* Michael Somos, Jun 26 2017 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 10 2016
STATUS
approved