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A349808 Number of cells in a regular 7-gon after n generations of mitosis.

%I A349808 #39 Jul 04 2023 14:23:22
%S A349808 1,50,190,400,680,1030,1450,1940,2500,3130,3830,4600,5440,6350,7330,
%T A349808 8380,9500,10690,11950,13280,14680,16150,17690,19300,20980,22730,
%U A349808 24550,26440,28400,30430,32530,34700,36940,39250,41630,44080,46600,49190,51850,54580,57380,60250,63190,66200,69280,72430,75650,78940,82300,85730,89230,92800
%N A349808 Number of cells in a regular 7-gon after n generations of mitosis.
%C A349808 See A350000 for background information.
%H A349808 Scott R. Shannon, <a href="/A349808/a349808.gif">Illustration for a(1) = 50</a>, the 7-gon after the first splitting.
%H A349808 Scott R. Shannon, <a href="/A349808/a349808_1.gif">Illustration for a(2) = 190</a>, the 7-gon after the second splitting.
%H A349808 Scott R. Shannon, <a href="/A349808/a349808_2.gif">Illustration for a(3) = 400</a> (third generation).
%H A349808 Scott R. Shannon, <a href="/A349808/a349808_3.gif">Illustration for a(4) = 680</a> (fourth generation).
%H A349808 Scott R. Shannon, <a href="/A349808/a349808_4.gif">Illustration for a(5) = 1030</a> (fifth generation).
%H A349808 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F A349808 a(0)=1; for n>=1, a(n) = 35*k^2+35*k-20.
%F A349808 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n >= 4, with a(0) = 1, a(1) = 50, a(2) = 190, a(3) = 400.
%F A349808 G.f.: (21*x^3-43*x^2-47*x-1)/(x-1)^3.
%F A349808 This sequence is most easily analyzed via the transition matrix M described in the Maple program.
%p A349808 with(LinearAlgebra):
%p A349808 M:=Matrix(5,5,[ [1,4,10,18,35], [0,0,0,6,7], [0,0,1,0,7], [0,0,0,0,0], [0,0,0,0,1]]);
%p A349808 v:=Matrix(5,1,[ [0], [0], [0], [0], [1]]); u:=Matrix(1,5,[1,1,1,1,1]);
%p A349808 A349808:=n->(u.M^n.v)[1,1];
%Y A349808 Row 7 of the array in A350000.
%K A349808 nonn
%O A349808 0,2
%A A349808 _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 06 2021