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 A247904 Start with a single pentagon; at n-th generation add a pentagon at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.) 7
 1, 6, 21, 56, 131, 286, 601, 1236, 2511, 5066, 10181, 20416, 40891, 81846, 163761, 327596, 655271, 1310626, 2621341, 5242776, 10485651, 20971406, 41942921, 83885956, 167772031, 335544186, 671088501, 1342177136, 2684354411, 5368708966, 10737418081 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Refer to A247619, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e."vertex to side" expansion version. Let us assign the label "1" to the pentagon at the origin; at n-th generation add a pentagon at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping pentagons will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The pentagons count is A005891. See illustration. For n >= 1, (a(n) - a(n-1))/5 is A000225. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Kival Ngaokrajang, Illustration of initial terms Index entries for linear recurrences with constant coefficients, signature (4,-5,2). FORMULA a(0) = 1, for n >= 1, a(n) = 5*A000225(n) + a(n-1). a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). - Colin Barker, Sep 26 2014 G.f.: (1+2*x+2*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Sep 26 2014 From G. C. Greubel, Feb 18 2022: (Start) a(n) = 10*2^n - (5*n + 9). E.g.f.: 10*exp(2*x) - (9 + 5*x)*exp(x). (End) MATHEMATICA LinearRecurrence[{4, -5, 2}, {1, 6, 21}, 51] (* G. C. Greubel, Feb 18 2022 *) PROG (PARI) a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1)) for (n=0, 50, print1(a(n), ", ")) (PARI) Vec(-(2*x^2+2*x+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014 (Magma) [10*2^n -(5*n+9): n in [0..50]]; // G. C. Greubel, Feb 18 2022 (Sage) [5*2^(n+1) -(5*n+9) for n in (0..50)] # G. C. Greubel, Feb 18 2022 CROSSREFS Cf. Vertex to vertex version: A061777, A247618, A247619, A247620. Cf. Vertex to side version: A101946, A247903, A247905. Cf. A000225, A005891. Sequence in context: A337895 A145134 A256571 * A074745 A296821 A056414 Adjacent sequences: A247901 A247902 A247903 * A247905 A247906 A247907 KEYWORD nonn,easy AUTHOR Kival Ngaokrajang, Sep 26 2014 EXTENSIONS More terms from Colin Barker, Sep 26 2014 STATUS approved

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Last modified February 21 10:38 EST 2024. Contains 370228 sequences. (Running on oeis4.)