A264788 a(n) is the number of circles added at n-th iteration of the pattern starting with 2 circles. (See comment.)

2, 2, 4, 4, 6, 10, 10, 12, 16, 16, 18, 22, 22, 24, 28, 28, 30, 34, 34, 36, 40, 40, 42, 46, 46, 48, 52, 52, 54, 58, 58, 60, 64, 64, 66, 70, 70, 72, 76, 76, 78, 82, 82, 84, 88, 88, 90, 94, 94, 96, 100, 100, 102, 106, 106, 108, 112, 112, 114, 118, 118, 120, 124

Offset: 0

Author

Kival Ngaokrajang, Nov 25 2015

Keywords

• nonn
• easy

Comments

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.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Kival Ngaokrajang, Illustration of initial terms
• Sacred Geometry, Flower of Life
• Eric Weisstein's World of Mathematics, Flower of Life
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A052380, A136289.

Programs

Mathematica

LinearRecurrence[{1,0,1,-1},{2,2,4,4,6,10},100] (* Paolo Xausa, Nov 17 2023 *)

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, ", "))}

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

Formula

From Colin Barker, Dec 10 2015: (Start)

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

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

(End)