cp's OEIS Frontend

A269965 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 A269965 #27 Jun 02 2025 12:16:54
%S A269965 1,3,10,26,63,145,332,760,1745,4007,9198,21102,48403,111021,254656,
%T A269965 584132,1339893,3073459,7049906,16171066,37093175,85084313,195166404,
%U A269965 447672720,1026871705,2355438303,5402904310,12393181766,28427480091,65206953349,149571708488
%N A269965 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 A269965 This is an auxiliary sequence, the main one being A269962.
%C A269965 a(n) is the number of squares colored red in the illustration.
%C A269965 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 A269965 Colin Barker, <a href="/A269965/b269965.txt">Table of n, a(n) for n = 5..1000</a>
%H A269965 Paolo Franchi, <a href="/A269965/a269965.pdf">Illustration of initial terms</a>
%H A269965 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,4,0,-2).
%F A269965 a(1)=a(2)=a(3)=a(4)=0, for n>= 5, a(n) = A269963(n-4)+a(n-1).
%F A269965 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + 2*a(n-4) + 2*a(n-5) + 5.
%F A269965 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 4*a(n-4) - 2*a(n-6).
%F A269965 G.f.: x^5*(1+2*x^2+2*x^3) / ((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)). - _Colin Barker_, Mar 09 2016
%t A269965 RecurrenceTable[{a[n + 1] ==
%t A269965    2 a[n] + a[n - 1] - 2 a[n - 2] + 2 a[n - 3] + 2 a[n - 4] + 5,
%t A269965   a[5] == 1, a[6] == 3, a[7] == 10, a[8] == 26, a[9] == 63}, a, {n, 5,
%t A269965    30}]
%t A269965 RecurrenceTable[{a[n + 1] ==
%t A269965    3 a[n] - a[n - 1] - 3 a[n - 2] + 4 a[n - 3] - 2 a[n - 5],
%t A269965   a[5] == 1, a[6] == 3, a[7] == 10, a[8] == 26, a[9] == 63,
%t A269965   a[10] == 145}, a, {n, 5, 30}]
%o A269965 (PARI) Vec(x^5*(1+2*x^2+2*x^3)/((1-x)*(1+x)*(1-3*x+2*x^2-2*x^4)) + O(x^50)) \\ _Colin Barker_, Mar 09 2016
%Y A269965 Main sequence: A269962.
%Y A269965 Other auxiliary sequences: A269963, A269964.
%K A269965 nonn,easy
%O A269965 5,2
%A A269965 _Paolo Franchi_, Mar 09 2016