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

%I A338200 #31 Dec 02 2021 13:32:09
%S A338200 0,0,1,2,4,6,9,12,17,21,27,33,41,48,58,67,79,90,104,117,134,149,168,
%T A338200 186,208,228,253,276,304,330,361,390,425,457,495,531,573,612,658,701,
%U A338200 751,798,852,903,962,1017,1080,1140,1208,1272,1345,1414,1492,1566,1649
%N A338200 The number of similarity classes of pointed reflection spaces of residue two in an n-dimensional vector space over GF(2).
%H A338200 Michael De Vlieger, <a href="/A338200/b338200.txt">Table of n, a(n) for n = 1..10000</a>
%H A338200 Saeid Azam, Masaya Tomie, and Yoji Yoshii, <a href="https://doi.org/10.18910/83200">Classification of pointed reflection spaces</a>, Osaka J. Math. (2021) Vol. 58, 563-589.
%H A338200 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-2,0,0,1,1,-1).
%F A338200 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.
%F A338200 From _Andrew Howroyd_, Oct 29 2020: (Start)
%F A338200 a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 10.
%F A338200 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)).
%F A338200 (End)
%t A338200 F[n_] := If[EvenQ[n],
%t A338200   n (n - 2)/8 +
%t A338200    2*Sum[Length[IntegerPartitions[k, {3}]], {k, 3, n/2}] +
%t A338200    Length[IntegerPartitions[(n + 2)/2, {3}]],
%t A338200   2*Floor[(n - 1)/4]*Floor[(n + 1)/4] +
%t A338200    2*Sum[Length[IntegerPartitions[k, {3}]], {k, 3, (n - 1)/2}] +
%t A338200    Length[IntegerPartitions[(n + 1)/2, {3}]] +
%t A338200    Length[IntegerPartitions[(n + 3)/2, {3}]]]
%t A338200 (* Second program: *)
%t A338200 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 *)
%o A338200 (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
%Y A338200 Cf. A069905.
%K A338200 nonn,easy
%O A338200 1,4
%A A338200 _Masaya Tomie_, Oct 16 2020