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A278181 Hexagonal spiral constructed on the nodes of the triangular net in which each new term is the sum of its neighbors.

%I A278181 #32 Nov 24 2016 09:42:04
%S A278181 1,1,2,3,4,5,7,8,9,12,14,19,22,29,33,42,47,59,74,82,99,108,129,155,
%T A278181 169,202,243,265,316,378,411,486,575,622,728,861,1017,1099,1280,1487,
%U A278181 1595,1832,2116,2440,2609,2980,3425,3933,4198,4779,5473,6262,6673,7570,8631,9828,10450,11800,13389,15267,17383
%N A278181 Hexagonal spiral constructed on the nodes of the triangular net in which each new term is the sum of its neighbors.
%C A278181 To evaluate a(n) consider the neighbors of a(n) that are present in the spiral when a(n) should be a new term in the spiral.
%H A278181 JungHwan Min, <a href="/A278181/b278181.txt">Table of n, a(n) for n = 0..10000</a>
%e A278181 Illustration of initial terms as a spiral:
%e A278181 .
%e A278181 .             22 - 19 - 14
%e A278181 .             /          \
%e A278181 .           29    3 - 2   12
%e A278181 .           /    /     \   \
%e A278181 .         33    4   1 - 1   9
%e A278181 .           \    \         /
%e A278181 .           42    5 - 7 - 8
%e A278181 .             \
%e A278181 .             47 - 59 - 74
%e A278181 .
%e A278181 a(16) = 47 because the sum of its two neighbors is 42 + 5 = 47.
%e A278181 a(17) = 59 because the sum of its three neighbors is 47 + 5 + 7 = 59.
%e A278181 a(18) = 74 because the sum of its three neighbors is 59 + 7 + 8 = 74.
%e A278181 a(19) = 82 because the sum of its two neighbors is 74 + 8 = 82.
%t A278181 A278181[0] = A278181[1] = 1; A278181[n_] := A278181[n] = With[{lev = Ceiling[1/6 (-3 + Sqrt[3] Sqrt[3 + 4 n])]}, With[{pos = 3 lev (lev - 1) + (n - 3 lev (1 + lev))/lev*(lev - 1)}, A278181[n - 1] + A278181[Ceiling[pos]] + If[Mod[n, lev] == 0 || n - 3 lev (lev - 1) == 1, 0, A278181[Floor[pos]]] + If[3 lev (1 + lev) == n, A278181[n - 6 lev + 1], 0]]]; Array[A278181, 61, 0] (* _JungHwan Min_, Nov 21 2016 *)
%Y A278181 Cf. A047931, A064642, A122479, A141481, A274821, A274921, A275606, A275610.
%K A278181 nonn
%O A278181 0,3
%A A278181 _Omar E. Pol_, Nov 14 2016