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A269962 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 at n-th stage.

%I A269962 #21 Mar 05 2022 01:29:22
%S A269962 1,5,17,45,105,237,537,1229,2825,6493,14905,34189,78409,179837,412505,
%T A269962 946221,2170473,4978653,11420025,26195213,60086537,137826493,
%U A269962 316146457,725176813,1663410601,3815531165,8752065209,20075486925,46049151561,105627543165
%N A269962 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 at n-th stage.
%C A269962 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 A269962 Colin Barker, <a href="/A269962/b269962.txt">Table of n, a(n) for n = 1..1000</a>
%H A269962 Paolo Franchi, <a href="/A269962/a269962.pdf">Illustration of initial terms</a>
%H A269962 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2,2,-2).
%F A269962 a(1)=1, for n>=1, a(n) = 2*A269963(n) + 2*A269963(n-1) - 1.
%F A269962 Linear non-homogeneous recurrence relation:
%F A269962 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5) + 4.
%F A269962 Linear homogeneous recurrence relation:
%F A269962 a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5).
%F A269962 G.f.: x*(1+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 A269962 RecurrenceTable[{a[n + 1] ==
%t A269962    4 a[n] - 5 a[n - 1] + 2 a[n - 2] + 2 a[n - 3] - 2 a[n - 4],
%t A269962   a[1] == 1, a[2] == 5, a[3] == 17, a[4] == 45, a[5] == 105}, a, {n,
%t A269962   1, 30}]
%t A269962 RecurrenceTable[{a[n + 1] ==
%t A269962    2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4] + 4,
%t A269962   a[1] == 1, a[2] == 5, a[3] == 17, a[4] == 45, a[5] == 105}, a, {n,
%t A269962   1, 30}]
%o A269962 (PARI) Vec(x*(1+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 A269962 Cf. A247618.
%Y A269962 Auxiliary sequences: A269963, A269964, A269965.
%K A269962 nonn,easy
%O A269962 1,2
%A A269962 _Paolo Franchi_, Mar 08 2016