This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334169 #29 Dec 20 2024 21:37:06
%S A334169 1,2,4,6,8,10,14,16,18,26,30,32,34,50,58,62,64,66,98,114,122,126,128,
%T A334169 130,194,226,242,250,254,256,258,386,450,482,498,506,510,512,514,770,
%U A334169 898,962,994,1010,1018,1022,1024,1026,1538,1794,1922,1986,2018,2034,2042,2046,2048,2050,3074,3586,3842
%N A334169 a(n) is the number of ON-cells in the n-th full level of ON-cells of a triangular wedge in the hexagonal grid of A151723 (after 2^k >= n generations have been computed).
%C A334169 a(n) also is the distance of the full level of ON-cells from the apex of the triangular wedge. Note that 7 is the last generation modifying level 6 and, more generally for example, generation 2^m + 2^(m-1) + 1 is the last generation modifying level 2^m + 2, for m >= 1:
%C A334169 Level Generation ON-cells
%C A334169 1 1 1
%C A334169 2 2 1 1
%C A334169 3 3 1 0 1
%C A334169 4 4 1 1 1 1
%C A334169 5 5 1 0 0 0 1
%C A334169 6 7 1 1 1 1 1 1
%C A334169 7 7 1 0 1 0 1 0 1
%C A334169 8 8 1 1 1 1 1 1 1 1
%C A334169 9 9 1 0 0 0 0 0 0 0 1
%C A334169 10 13 1 1 1 1 1 1 1 1 1 1
%C A334169 ...
%H A334169 Hartmut F. W. Hoft, <a href="/A334169/a334169.pdf">Proof of positions of full levels</a>
%F A334169 a(0) = 1; a(1) = 2, a(n) = 2 + Sum_{i=0..j} 2^(k-i-1), where k = floor((3 + sqrt(1 + 8*(n-2)))/2) and j = n - 2 - (k-2)*(k-1)/2 for n >= 2.
%e A334169 The sequence is the triangle below read by rows, where each row contains m-1 full levels of ON-cells from level 2^(m-1) + 2 through level 2^m, for m >= 2:
%e A334169 m\j 0 1 2 3 4 5 6 7 8
%e A334169 0: 1
%e A334169 1: 2
%e A334169 2: 4
%e A334169 3: 6 8
%e A334169 4: 10 14 16
%e A334169 5: 18 26 30 32
%e A334169 6: 34 50 58 62 64
%e A334169 7: 66 98 114 122 126 128
%e A334169 8: 130 194 226 242 250 254 256
%e A334169 9: 258 386 450 482 498 506 510 512
%e A334169 10: 514 770 898 962 994 1010 1018 1022 1024
%e A334169 ...
%e A334169 A formula for the m-1 elements in positions (m, j), 0 <= j <= m-2, in each row m >= 2 is: b(m, j) = 2 + Sum_{k=0..j} 2^(m-k-1).
%t A334169 triangleRow[m_] := Map[2+Sum[2^(m-k-1), {k, 0, #}]&, Range[0, m-2]]/;m>=2
%t A334169 triangleRow[10] (* last line in triangle in Comments section *)
%t A334169 a334169[0]=1; a334169[1]=2; a334169[n_] := Module[{k, j}, k=Floor[(3 + Sqrt[1 + 8(n-2)])/2]; j = n - 2 - (k-2)(k-1)/2; 2 + Sum[2^(k-i-1), {i, 0, j}]]/;n>=2
%t A334169 Map[a334169,Range[0,66]] (* sequence data *)
%Y A334169 Cf. A151723.
%K A334169 nonn,tabf
%O A334169 0,2
%A A334169 _Hartmut F. W. Hoft_, Apr 17 2020