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Van Eck sequence in 2-dimensional hexagonal space.
2

%I #32 Aug 03 2024 17:10:52

%S 0,0,1,0,1,2,0,1,2,3,0,2,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N Van Eck sequence in 2-dimensional hexagonal space.

%C Fill a board made from hexagonal cells with numbers using the following rules:

%C - write a 0 in the starting cell;

%C - if the number just written had not previously been on the board then the next number is 0;

%C - otherwise, the next number is the distance from its closest occurrence, counting cells you need to pass through to reach it.

%C ______ ______

%C / \ / \

%C / \ / \

%C ______/ \______/ \______

%C / \ / \ / \

%C / \ / \ / \

%C / \______/ 2 \______/ \

%C \ / \ . . / \ /

%C \ / .\ / . \ /

%C \______/ 1 \______/ 3 \______/

%C / \ . / \ . / \

%C / \ . / \ / . \

%C / 1 \___.__/ 0 \______/ 0 \

%C \ . / . \ . . / \ . /

%C \ . / . \ . / . \ . /

%C \___.__/ 0 \___.__/ 1 \___.__/

%C / . \ . / ^ \ . / . \

%C / . \ . / | \ . / . \

%C / 1 \___.__/ 0 \___.__/ 2 \

%C \ . / . \ START / . \ . /

%C \ . / . \ / . \ . /

%C \___.__/ 2 \______/ 0 \___.__/

%C / . \ . / \ . / . \

%C / . \ / . .\ / . \

%C / 1 \______/ 1 \______/ 3 \

%C \ . / \ / \ . /

%C \ / . \ / .\ /

%C \______/ 1 \______/ 3 \______/

%C / \ . / \ . / \

%C / \ / . .\ / \

%C / \______/ 1 \______/ \

%C \ / \ / \ /

%C \ / \ / \ /

%C \______/ \______/ \______/

%C .

%C a(n) = 1 for all n > 15, because the previous 1 will always be adjacent to another 1. - _Charlie Neder_, Jun 11 2019

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F G.f.: x^3*(1 - x + x^2 + x^3 - 2*x^4 + x^5 + x^6 + x^7 - 3*x^8 + 2*x^9 + x^10 - 2*x^12)/(1 - x). - _Elmo R. Oliveira_, Aug 03 2024

%Y Cf. A181391, A308626.

%K nonn,easy

%O 1,6

%A _Jacek Sandomierz_, Jun 11 2019

%E Extended by _Charlie Neder_, Jun 13 2019