A264788 - OEIS

%I #19 Nov 17 2023 11:20:42

%S 2,2,4,4,6,10,10,12,16,16,18,22,22,24,28,28,30,34,34,36,40,40,42,46,

%T 46,48,52,52,54,58,58,60,64,64,66,70,70,72,76,76,78,82,82,84,88,88,90,

%U 94,94,96,100,100,102,106,106,108,112,112,114,118,118,120,124

%N a(n) is the number of circles added at n-th iteration of the pattern starting with 2 circles. (See comment.)

%C Pattern construction rules: (i) At n = 0, there are two circles of radius s with centers at the ends of a straight line of length s. (ii) At n > 0, draw circles by placing center at the intersection points of the circumferences of circles in the previous iteration, with overlaps forbidden. The pattern seems to be the flower of life. See illustration.

%H Colin Barker, <a href="/A264788/b264788.txt">Table of n, a(n) for n = 0..1000</a>

%H Kival Ngaokrajang, <a href="/A264788/a264788.pdf">Illustration of initial terms</a>

%H Sacred Geometry, <a href="http://www.bibliotecapleyades.net/geometria_sagrada/esp_geometria_sagrada_6.htm">Flower of Life</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FlowerofLife.html">Flower of Life</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F From _Colin Barker_, Dec 10 2015: (Start)

%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.

%F G.f.: 2*(1+x^2-x^3+x^4+x^5) / ((1-x)^2*(1+x+x^2)).

%F (End)

%t LinearRecurrence[{1,0,1,-1},{2,2,4,4,6,10},100] (* _Paolo Xausa_, Nov 17 2023 *)

%o (PARI) {a = 4; print1("2, 2, ", a, ", "); for(n = 2, 100, if (Mod(n,3)==0, d1 = 2); if (Mod(n,3)==1, d1 = 4);  if (Mod(n,3)==2, d1 = 0); a = a + d1; print1(a, ", "))}

%o (PARI) Vec(2*(1+x^2-x^3+x^4+x^5)/((1-x)^2*(1+x+x^2)) + O(x^100)) \\ _Colin Barker_, Dec 10 2015

%Y Cf. A052380, A136289.

%K nonn,easy

%O 0,1

%A _Kival Ngaokrajang_, Nov 25 2015