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A338200
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The number of similarity classes of pointed reflection spaces of residue two in an n-dimensional vector space over GF(2).
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1
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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
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OFFSET
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1,4
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).
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FORMULA
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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.
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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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