A091573 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_6.

7, 12, 17, 24, 31, 36, 41, 48, 55, 60, 65, 72, 79, 84, 89, 96, 103, 108, 113, 120, 127, 132, 137, 144, 151, 156, 161, 168, 175, 180, 185, 192, 199, 204, 209, 216, 223, 228, 233, 240, 247, 252, 257, 264, 271, 276, 281, 288, 295, 300, 305, 312, 319, 324, 329

Offset: 0

Author

Paul Boddington, Jan 22 2004

Keywords

• easy
• nonn

References

• I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, Vol. 44, Number 5.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A004525, A091571, A091572, A091574, A091575, A091576, A091577.

Programs

Mathematica

CoefficientList[ Series[ (7 - 2x + 7x^2) / (1 - 2x + 2x^2 - 2x^3 + x^4), {x, 0, 49}], x] (* Jean-François Alcover, Dec 02 2011 *)

PARI

a(n) = (12+(-I)^n+I^n+12*n)/2 \\ Colin Barker, Oct 18 2015

PARI

Vec((7-2*x+7*x^2)/((1+x^2)*(1-x)^2) + O(x^100)) \\ Colin Barker, Oct 18 2015

PARI

a(n) = if(n%2 == 1, 6*n+6, if(n%4 == 0, 6*n+7, 6*n+5));
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 18 2015

Formula

a(n) = 6*n+6 (n odd), 6*n+7 (n==0 (mod 4)), 6*n+5 (n==2 (mod 4)).

G.f.: (7-2*x+7*x^2) / ((1+x^2)*(1-x)^2).

From Colin Barker, Oct 18 2015: (Start)

a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>3.

a(n) = (12+(-i)^n+i^n+12*n)/2 where i = sqrt(-1).

(End)

Extensions

G.f. corrected by Colin Barker, Oct 18 2015