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

0,1

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).

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)

Mathematica

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

Prog

(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

Crossrefs

Cf. A052380, A136289.

Sequence in context: A325253 A143483 A323093 * A192326 A131733 A128745

Adjacent sequences: A264785 A264786 A264787 * A264789 A264790 A264791

Keyword

nonn,easy

Author

Kival Ngaokrajang, Nov 25 2015

Status

approved