A338200 The number of similarity classes of pointed reflection spaces of residue two in an n-dimensional vector space over GF(2).

0, 0, 1, 2, 4, 6, 9, 12, 17, 21, 27, 33, 41, 48, 58, 67, 79, 90, 104, 117, 134, 149, 168, 186, 208, 228, 253, 276, 304, 330, 361, 390, 425, 457, 495, 531, 573, 612, 658, 701, 751, 798, 852, 903, 962, 1017, 1080, 1140, 1208, 1272, 1345, 1414, 1492, 1566, 1649

Offset: 1

Author

Masaya Tomie, Oct 16 2020

Keywords

• nonn
• easy

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Saeid Azam, Masaya Tomie, and Yoji Yoshii, Classification of pointed reflection spaces, Osaka J. Math. (2021) Vol. 58, 563-589.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Crossrefs

Cf. A069905.

Programs

Mathematica

F[n_] := If[EvenQ[n],
  n (n - 2)/8 +
   2*Sum[Length[IntegerPartitions[k, {3}]], {k, 3, n/2}] +
   Length[IntegerPartitions[(n + 2)/2, {3}]],
  2*Floor[(n - 1)/4]*Floor[(n + 1)/4] +
   2*Sum[Length[IntegerPartitions[k, {3}]], {k, 3, (n - 1)/2}] +
   Length[IntegerPartitions[(n + 1)/2, {3}]] +
   Length[IntegerPartitions[(n + 3)/2, {3}]]]
(* Second program: *)
LinearRecurrence[{1,1,0,0,-2,0,0,1,1,-1}, {0,0,1,2,4,6,9,12,17,21}, 55] (* Jean-François Alcover, Nov 13 2020 *)

PARI

concat([0,0], Vec((1 + x + x^2 - x^4 - x^5)/((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^50))) \\ Andrew Howroyd, Oct 29 2020

Formula

a(n) = (1/8)*n*(n-2) + 2*(Sum_{k=3..n/2} p(k,3)) + p((n+2)/2,3) if n is even; a(n) = 2*floor((n-1)/4)*floor((n+1)/4) + 2*(Sum_{k=3..(n-1)/2} p(k,3)) + p((n+1)/2,3) + p((n+3)/2,3) if n is odd, where p(k,3) = A069905(k) is the number of partitions of k into three parts.

From Andrew Howroyd, Oct 29 2020: (Start)

a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 10.

G.f.: x^3*(1 + x + x^2 - x^4 - x^5)/((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).

(End)