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A269964 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 A269964 #21 Mar 05 2022 01:30:52
%S A269964 1,1,3,5,11,23,53,121,279,639,1465,3357,7699,17659,40509,92921,213143,
%T A269964 488903,1121441,2572357,5900475,13534515,31045477,71212113,163346335,
%U A269964 374683807,859449705,1971405725,4522010435,10372587467,23792640941,54575559337
%N A269964 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 A269964 This is an auxiliary sequence, the main one being A269962.
%C A269964 a(n) gives the number of squares colored red in the illustration.
%C A269964 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 A269964 Colin Barker, <a href="/A269964/b269964.txt">Table of n, a(n) for n = 1..1000</a>
%H A269964 Paolo Franchi, <a href="/A269964/a269964.pdf">Illustration of initial terms</a>
%H A269964 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,4,0,-2).
%F A269964 a(n) = 2*a(n-2) + 2*a(n-3) + 2*A269965(n) + 1.
%F A269964 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5) - 2.
%F A269964 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 4*a(n-4) - 2*a(n-6).
%F A269964 G.f.: x*(1-2*x+x^2-2*x^4) / ((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)). - _Colin Barker_, Mar 09 2016
%t A269964 RecurrenceTable[{a[n + 1] ==
%t A269964    2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4] - 2,
%t A269964   a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 5, a[5] == 11}, a, {n, 1,
%t A269964   30}]
%t A269964 RecurrenceTable[{a[n + 1] ==
%t A269964    3 a[n] - a[n - 1] - 3 a[n - 2] + 4 a[n - 3] - 2 a[n - 5],
%t A269964   a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 5, a[5] == 11,
%t A269964   a[6] == 23}, a, {n, 1, 30}]
%o A269964 (PARI) Vec(x*(1-2*x+x^2-2*x^4)/((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)) + O(x^50)) \\ _Colin Barker_, Mar 09 2016
%Y A269964 Main sequence: A269962.
%Y A269964 Other auxiliary sequences: A269963, A269965.
%K A269964 nonn,easy
%O A269964 1,3
%A A269964 _Paolo Franchi_, Mar 09 2016