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A263135 The maximum number of penny-to-penny connections when n pennies are placed on the vertices of a hexagonal tiling.

%I A263135 #21 Nov 23 2015 01:44:09
%S A263135 0,0,1,2,3,4,6,7,8,9,11,12,13,15,16,17,19,20,21,23,24,25,27,28,30,31,
%T A263135 32,34,35,36,38,39,41,42,43,45,46,48,49,50,52,53,55,56,57,59,60,62,63,
%U A263135 64,66,67,69,70,72,73,74,76,77,79,80,81,83,84,86,87,89,90
%N A263135 The maximum number of penny-to-penny connections when n pennies are placed on the vertices of a hexagonal tiling.
%C A263135 a(A033581(n)) = A152743(n).
%C A263135 1 <= a(n+1) - a(n) <=2 for all n > 0.
%C A263135 Lim_{n -> infinity} a(n)/n = 3/2.
%C A263135 Conjecture: a(2*n) - A047932(n) = A216256(n) for n > 0.
%H A263135 Peter Kagey, <a href="/A263135/b263135.txt">Table of n, a(n) for n = 0..10000</a>
%e A263135 .           |            |     o o     .
%e A263135 .           |      o o   |  o o   o o  .
%e A263135 .    o o    |   o o   o  | o   o o   o .
%e A263135 .   o   o   |  o   o o   |  o o   o o  .
%e A263135 .    o o    |   o o      | o   o o   o .
%e A263135 .           |            |  o o   o o  .
%e A263135 .           |            |     o o     .
%e A263135 .           |            |             .
%e A263135 . f(6) = 6  | f(10) = 11 | f(24) = 30  .
%Y A263135 Cf. A047932 (triangular tiling), A123663 (square tiling).
%Y A263135 Cf. A033581, A152743.
%K A263135 nonn
%O A263135 0,4
%A A263135 _Peter Kagey_, Oct 10 2015