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 A317303 #54 Jun 17 2021 17:46:20 %S A317303 2,7,8,9,16,17,18,19,20,29,30,31,32,33,34,35,46,47,48,49,50,51,52,53, %T A317303 54,67,68,69,70,71,72,73,74,75,76,77,92,93,94,95,96,97,98,99,100,101, %U A317303 102,103,104,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,154,155,156,157,158,159,160 %N A317303 Numbers k such that both Dyck paths of the symmetric representation of sigma(k) have a central peak. %C A317303 Also triangle read by rows which gives the odd-indexed rows of triangle A014132. %C A317303 There are no triangular number (A000217) in this sequence. %C A317303 For more information about the symmetric representation of sigma see A237593 and its related sequences. %C A317303 Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an odd number of peaks. - _Omar E. Pol_, Sep 13 2018 %e A317303 Written as an irregular triangle in which the row lengths are the odd numbers, the sequence begins: %e A317303 2; %e A317303 7, 8, 9; %e A317303 16, 17, 18, 19, 20; %e A317303 29, 30, 31, 32, 33, 34, 35; %e A317303 46, 47, 48, 49, 50, 51, 52, 53, 54; %e A317303 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77; %e A317303 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104; %e A317303 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135; %e A317303 ... %e A317303 Illustration of initial terms: %e A317303 ----------------------------------------------------------- %e A317303 k sigma(k) Diagram of the symmetry of sigma %e A317303 ----------------------------------------------------------- %e A317303 _ _ _ _ _ _ _ _ _ %e A317303 _| | | | | | | | | | | | %e A317303 2 3 |_ _| | | | | | | | | | | %e A317303 | | | | | | | | | | %e A317303 _|_| | | | | | | | | %e A317303 _| _ _|_| | | | | | | %e A317303 _ _ _ _| _| | | | | | | | %e A317303 7 8 |_ _ _ _| |_ _| | | | | | | %e A317303 8 15 |_ _ _ _ _| _ _ _| | | | | | %e A317303 9 13 |_ _ _ _ _| | _ _ _|_| | | | %e A317303 _| | _ _ _|_| | %e A317303 _| _| | _ _ _ _| %e A317303 _ _| _| _ _| | %e A317303 | _ _| _| _| %e A317303 | | | | %e A317303 _ _ _ _ _ _ _ _| | _ _| _ _| %e A317303 16 31 |_ _ _ _ _ _ _ _ _| | _ _| %e A317303 17 18 |_ _ _ _ _ _ _ _ _| | | %e A317303 18 39 |_ _ _ _ _ _ _ _ _ _| | %e A317303 19 20 |_ _ _ _ _ _ _ _ _ _| | %e A317303 20 42 |_ _ _ _ _ _ _ _ _ _ _| %e A317303 . %e A317303 For the first nine terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central peak. %e A317303 Compare with A317304. %Y A317303 Column 1 gives A130883, n >= 1. %Y A317303 Column 2 gives A033816, n >= 1. %Y A317303 Row sums give the odd-indexed terms of A006002. %Y A317303 Right border gives the positive terms of A014107, also the odd-indexed terms of A000096. %Y A317303 The union of A000217, A317304 and this sequence gives A001477. %Y A317303 Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317304. See also A317306. %Y A317303 Cf. A000203, A005408, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626. %K A317303 nonn,tabf %O A317303 1,1 %A A317303 _Omar E. Pol_, Aug 27 2018