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The generating function of the rationals A060819(n)/a(n) = 1/2 - 1/(n+2), n >= 0, with A060819(0) = 0, mentioned in the comment on a sum by Gary Detlefs above is (1/2)*(1-hypergeom([1, 1], [3], -x/(1-x)))/(1-x) = (x*(2 - x) + 2*(1 - x)*log(1-x) )/(2*(1-x)*x^2). Thanks to him for leading me to Jolley's general remark (201) on p. 38 on such sums. - Wolfdieter Lang, Mar 08 2018
The generating function of the rationals A060819(n)/a(n) = 1/2 - 1/(n+2), n >= 0, with A060819(0) = 0, mentioned in the comment on a sum by Gary Detlefs above is (1/2)*(1-hypergeom([1, 1], [3], -x/(1-x)))/(1-x) = (x*(2 - x) + 2*(1 - x)*log(1-x) )/(2*(1-x)*x^2). Thanks to him for leading me to Jolley's general remark (201) on p. 38 on such sums. - Wolfdieter Lang, Mar 08 2018
The above b(n) also relates rotoinversions (rotation + inversion through the origin) to rotoreflections (rotation + reflection in a plane normal to the rotation axis). An n-fold rotoinversion clockwise is the same as some number of b(n)-fold rotoreflections counterclockwise. The Schoenflies notation for point group symmetry common in chemistry describes improper rotations as rotoreflections, while the International (Hemann-Mauguin) notation favored in crystallography describes them as rotoinversions. - R. James Evans, Nov. 6 06 2024
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The above b(n) also relates rotoinversions (rotation + inversion through the origin) to rotoreflections (rotation + reflection in a plane normal to the rotation axis). An n-fold rotoinversion clockwise is the same as some number of b(n)-fold rotoreflections counterclockwise. The Schoenflies notation for point group symmetry common in chemistry describes improper rotations as rotoreflections, while the International (Hemann-Mauguin) notation favoured favored in crystallography describes them as rotoinversions. - R. James Evans, Nov. 6 2024
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The above b(n) also relates rotoinversions (rotation + inversion through the origin) to rotoreflections (rotation + reflection in a plane normal to the rotation axis). An n-fold rotoinversion clockwise is the same as some number of b(n)-fold rotoreflections counterclockwise. The Schoenflies notation for point group symmetry common in chemistry describes improper rotations as rotoreflections, while the International (Hemann-Mauguin) notation favoured in crystallography describes them as rotoinversions. - R. James Evans, Nov. 6 2024
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