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

1,4

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).

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)

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 *)

Prog

(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

Crossrefs

Cf. A069905.

Sequence in context: A048171 A090178 A080548 * A194450 A080556 A229093

Adjacent sequences: A338197 A338198 A338199 * A338201 A338202 A338203

Keyword

nonn,easy

Author

Masaya Tomie, Oct 16 2020

Status

approved