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 A239660 #87 Apr 23 2023 07:24:00 %S A239660 1,1,1,1,2,2,2,2,2,1,1,2,2,1,1,2,3,1,1,3,3,1,1,3,3,2,2,3,3,2,2,3,4,1, %T A239660 1,1,1,4,4,1,1,1,1,4,4,2,1,1,2,4,4,2,1,1,2,4,5,2,1,1,2,5,5,2,1,1,2,5, %U A239660 5,2,2,2,2,5,5,2,2,2,2,5,6,2,1,1,1,1,2,6,6,2,1,1,1,1,2,6,6,3,1,1,1,1,3,6,6,3,1,1,1,1,3,6 %N A239660 Triangle read by rows in which row n lists two copies of the n-th row of triangle A237593. %C A239660 For the construction of this sequence also we can start from A235791. %C A239660 This sequence can be interpreted as an infinite Dyck path: UDUDUUDD... %C A239660 Also we use this sequence for the construction of a spiral in which the arms in the quadrants give the symmetric representation of sigma, see example. %C A239660 We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - _Omar E. Pol_, Dec 07 2016 %C A239660 The spiral has the property that the sum of the parts in the quadrants 1 and 3, divided by the sum of the parts in the quadrants 2 and 4, converges to 3/5. - _Omar E. Pol_, Jun 10 2019 %H A239660 Robert Price, <a href="/A239660/b239660.txt">Table of n, a(n) for n = 1..30016</a> (rows n = 1..412, flattened) %e A239660 Triangle begins (first 15.5 rows): %e A239660 1, 1, 1, 1; %e A239660 2, 2, 2, 2; %e A239660 2, 1, 1, 2, 2, 1, 1, 2; %e A239660 3, 1, 1, 3, 3, 1, 1, 3; %e A239660 3, 2, 2, 3, 3, 2, 2, 3; %e A239660 4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4; %e A239660 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4; %e A239660 5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5; %e A239660 5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5; %e A239660 6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6; %e A239660 6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6; %e A239660 7, 2, 2, 1, 1, 2, 2, 7, 7, 2, 2, 1, 1, 2, 2, 7; %e A239660 7, 3, 2, 1, 1, 2, 3, 7, 7, 3, 2, 1, 1, 2, 3, 7; %e A239660 8, 3, 1, 2, 2, 1, 3, 8, 8, 3, 1, 2, 2, 1, 3, 8; %e A239660 8, 3, 2, 1, 1, 1, 1, 2, 3, 8, 8, 3, 2, 1, 1, 1, 1, 2, 3, 8; %e A239660 9, 3, 2, 1, 1, 1, 1, 2, 3, 9, ... %e A239660 . %e A239660 Illustration of initial terms as an infinite Dyck path (row n = 1..4): %e A239660 . %e A239660 . /\/\ /\/\ %e A239660 . /\ /\ /\/\ /\/\ / \ / \ %e A239660 . /\/\/ \/ \/ \/ \/ \/ \ %e A239660 . %e A239660 . %e A239660 Illustration of initial terms for the construction of a spiral related to sigma: %e A239660 . %e A239660 . row 1 row 2 row 3 row 4 %e A239660 . _ _ _ %e A239660 . |_ %e A239660 . _ _ | %e A239660 . _ _ | | %e A239660 . | | | | %e A239660 . | | | | %e A239660 . |_ _ |_ _| | %e A239660 . |_ _ _ _| _| %e A239660 . _ _ _| %e A239660 . %e A239660 .[1,1,1,1] [2,2,2,2] [2,1,1,2,2,1,1,2] [3,1,1,3,3,1,1,3] %e A239660 . %e A239660 The first 2*A003056(n) terms of the n-th row are represented in the A010883(n-1) quadrant and the last 2*A003056(n) terms of the n-th row are represented in the A010883(n) quadrant. %e A239660 . %e A239660 Illustration of the spiral constructed with the first 15.5 rows of triangle: %e A239660 . %e A239660 . 12 _ _ _ _ _ _ _ _ %e A239660 . | _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7 %e A239660 . | | |_ _ _ _ _ _ _| %e A239660 . _| | | %e A239660 . |_ _|9 _ _ _ _ _ _ |_ _ %e A239660 . 12 _ _| | _ _ _ _ _|_ _ _ _ _ 5 |_ %e A239660 . _ _ _| | _| | |_ _ _ _ _| | %e A239660 . | _ _ _| 9 _|_ _| |_ _ 3 |_ _ _ 7 %e A239660 . | | _ _| | 12 _ _ _ _ |_ | | | %e A239660 . | | | _ _| _| _ _ _|_ _ _ 3 |_|_ _ 5 | | %e A239660 . | | | | _| | |_ _ _| | | | | %e A239660 . | | | | | _ _| |_ _ 3 | | | | %e A239660 . | | | | | | 3 _ _ | | | | | | %e A239660 . | | | | | | | _|_ 1 | | | | | | %e A239660 . _|_| _|_| _|_| _|_| |_| _|_| _|_| _|_| _ %e A239660 . | | | | | | | | | | | | | | | | %e A239660 . | | | | | | |_|_ _ _| | | | | | | | %e A239660 . | | | | | | 2 |_ _|_ _| _| | | | | | | %e A239660 . | | | | |_|_ 2 |_ _ _| _ _| | | | | | %e A239660 . | | | | 4 |_ 7 _| _ _| | | | | %e A239660 . | | |_|_ _ |_ _ _ _ | _| _ _ _| | | | %e A239660 . | | 6 |_ |_ _ _ _|_ _ _ _| | _| _ _| | | %e A239660 . |_|_ _ _ |_ 4 |_ _ _ _ _| _| | _ _ _| | %e A239660 . 8 | |_ _ | 15| _| | _ _ _| %e A239660 . |_ | |_ _ _ _ _ _ | _ _| _| | %e A239660 . 8 |_ |_ |_ _ _ _ _ _|_ _ _ _ _ _| | _| _| %e A239660 . |_ _| 6 |_ _ _ _ _ _ _| _ _| _| %e A239660 . | 28| _ _| %e A239660 . |_ _ _ _ _ _ _ _ | | %e A239660 . |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| | %e A239660 . 8 |_ _ _ _ _ _ _ _ _| %e A239660 . 31 %e A239660 . %e A239660 The diagram contains A237590(16) = 27 parts. %e A239660 The total area (also the total number of cells) in the n-th arm of the spiral is equal to sigma(n) = A000203(n), considering every quadrant and the axes x and y. (checked by hand up to row n = 128). The parts of the spiral are in A237270: 1, 3, 2, 2, 7... %e A239660 Diagram extended by _Omar E. Pol_, Aug 23 2018 %Y A239660 Row n has length 4*A003056(n). %Y A239660 The sum of row n is equal to 4*n = A008586(n). %Y A239660 Row n is a palindromic composition of 4*n = A008586(n). %Y A239660 Both column 1 and right border are A008619, n >= 1. %Y A239660 The connection between A196020 and A237270 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> this sequence --> A237270. %Y A239660 Cf. A000203, A000217, A003056, A008619, A010883, A112610, A193553, A237048, A237271, A237590, A239052, A239053, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A262626, A296508, A299778. %K A239660 nonn,look,tabf %O A239660 1,5 %A A239660 _Omar E. Pol_, Mar 24 2014