cp's OEIS Frontend

A269963 Start with a square; at each stage add a square at each expandable vertex so that the ratio of the side of the squares at stage n+1 and at stage n is the golden ratio phi=0.618...; a(n) is the number of squares in a portion of the n-th stage (see below).

%I A269963 #13 Mar 05 2022 01:30:52
%S A269963 1,2,7,16,37,82,187,428,985,2262,5191,11904,27301,62618,143635,329476,
%T A269963 755761,1733566,3976447,9121160,20922109,47991138,110082091,252506316,
%U A269963 579198985,1328566598,3047466007,6990277456,16034298325,36779473258,84364755139
%N A269963 Start with a square; at each stage add a square at each expandable vertex so that the ratio of the side of the squares at stage n+1 and at stage n is the golden ratio phi=0.618...; a(n) is the number of squares in a portion of the n-th stage (see below).
%C A269963 This is an auxiliary sequence, the main one being A269962.
%C A269963 a(n) gives the number of squares colored red in the illustration.
%C A269963 The ratio phi=0.618... is chosen so that from the fourth stage on some squares overlap perfectly. The figure displays some kind of fractal behavior. See illustration.
%H A269963 Colin Barker, <a href="/A269963/b269963.txt">Table of n, a(n) for n = 1..1000</a>
%H A269963 Paolo Franchi, <a href="/A269963/a269963.pdf">Illustration of initial terms</a>
%H A269963 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,2,2).
%F A269963 a(n) = 2*A269964(n) + a(n-1) - 1.
%F A269963 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5).
%F A269963 G.f.: x*(1+2*x^2+2*x^3) / ((1+x)*(1-3*x+2*x^2-2*x^4)). - _Colin Barker_, Mar 09 2016
%t A269963 RecurrenceTable[{a[n + 1] ==
%t A269963    2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4],
%t A269963   a[1] == 1, a[2] == 2, a[3] == 7, a[4] == 16, a[5] == 37}, a, {n, 1,
%t A269963   30}]
%o A269963 (PARI) Vec(x*(1+2*x^2+2*x^3)/((1+x)*(1-3*x+2*x^2-2*x^4)) + O(x^50)) \\ _Colin Barker_, Mar 09 2016
%Y A269963 Main sequence: A269962.
%Y A269963 Other auxiliary sequences: A269964, A269965.
%K A269963 nonn,easy
%O A269963 1,2
%A A269963 _Paolo Franchi_, Mar 09 2016