

A264788


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


3



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
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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



FORMULA

a(n) = a(n1) + a(n3)  a(n4) for n>5.
G.f.: 2*(1+x^2x^3+x^4+x^5) / ((1x)^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^2x^3+x^4+x^5)/((1x)^2*(1+x+x^2)) + O(x^100)) \\ Colin Barker, Dec 10 2015


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



