A014695 Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of Q_8.

1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

From Klaus Brockhaus, May 14 2010: (Start)

Periodic sequence: Repeat 1, 2, 2, 1.

a(n) = A130658(n+1).

Continued fraction expansion of (5+sqrt(221))/14.

Decimal expansion of 37/303. (End)

Links

• A. Adem, Recent developments in the cohomology of finite groups, Notices Amer. Math. Soc., 44 (1997), 806-812.
• Eric Weisstein's World of Mathematics, Simple Graph
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Crossrefs

Denominators for the sequence whose numerators are A064038.

Cf. A130658, A177841. - Klaus Brockhaus, May 14 2010

Programs

Mathematica

Table[Denominator[n*(n + 1)/4], {n, 0, 104}] (* Arkadiusz Wesolowski, Aug 09 2012 *)
LinearRecurrence[{1,-1,1},{1,2,2},120] (* Harvey P. Dale, Jan 19 2020 *)

PARI

x='x+O('x^100); Vec((1+2*x+2*x^2+x^3)/(1-x^4)) \\ Altug Alkan, Dec 24 2015

Python

def A014695(n): return (1,2,2,1)[n&3] # Chai Wah Wu, Apr 17 2023

Formula

G.f.: (1+x+x^2)/((1-x)*(1+x^2)) = (1+2*x+2*x^2+x^3)/(1-x^4).

a(n) = (3-sqrt(2)*cos((2*n+1)*Pi/4))/2. - Jaume Oliver Lafont, Nov 28 2009

a(n) = (6-(1+i)*i^n-(1-i)*(-i)^n)/4 where i = sqrt(-1). - Klaus Brockhaus, May 14 2010

a(n) = denominator of Sum_{k=0..n} k/2. - Arkadiusz Wesolowski, Aug 09 2012

Extensions

More terms from Klaus Brockhaus, May 14 2010