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 A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line. 4
 0, 3, 30, 189, 684, 1815, 3978, 7665, 13464, 22059, 34230, 50853, 72900, 101439, 137634, 182745, 238128, 305235, 385614, 480909, 592860, 723303, 874170, 1047489, 1245384, 1470075, 1723878, 2009205, 2328564, 2684559, 3079890, 3517353, 3999840, 4530339, 5111934, 5747805, 6441228, 7195575 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequences gives the number of polygon edges formed when connecting every pair of vertices of a regular polygon, with an odd number of vertices, by an infinite line. A bisection of A344899. - N. J. A. Sloane, Sep 12 2021 See A344857 for other examples and images of the polygons. LINKS Table of n, a(n) for n=1..38. Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = 4*n^4 - 22*n^3 + 44*n^2 - 35*n + 9 (see Sidorenko link in A344857 for proof). From Stefano Spezia, Jun 10 2021: (Start) G.f.: 3*x^2*(1 + 5*x + 23*x^2 + 3*x^3)/(1 - x)^5. a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End) EXAMPLE a(3) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer unbounded regions are also formed. PROG (Python) def A344907(n): return n*(n*(n*(4*n - 22) + 44) - 35) + 9 # Chai Wah Wu, Sep 12 2021 CROSSREFS Cf. A344899 (number of edges for all n-gons), A344866 (number of polygon), A146212, A344857, A344311, A007678, A331450, A344938. See also A347322. Sequence in context: A013220 A177727 A132413 * A032263 A003771 A121100 Adjacent sequences: A344904 A344905 A344906 * A344908 A344909 A344910 KEYWORD nonn,easy AUTHOR Scott R. Shannon, Jun 02 2021 STATUS approved

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Last modified June 16 02:19 EDT 2024. Contains 373416 sequences. (Running on oeis4.)