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A279286 a(n) = d if the point (d,d) is shared by a record of different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

%I A279286 #69 Feb 09 2017 10:25:21
%S A279286 1,2,7,15,52,102,296,371,455,929,1853,2034,4517,4797,5829,6146,6948,
%T A279286 17577,18915,60349,78369,85171,123788,128596,415355,906771,1308771,
%U A279286 3329668
%N A279286 a(n) = d if the point (d,d) is shared by a record of different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.
%C A279286 Is this sequence infinite?
%C A279286 First differs from A282197 (another version) at a(19). - _Omar E. Pol_, Feb 08 2017
%C A279286 a(n) = d if the point (d,d) belongs to a vertical-line-segment whose length is a record in the main diagonal of the pyramid described in A245092 (starting from the top). The diagram of the symmetries of sigma is also the top view of the mentioned pyramid. See examples. - _Omar E. Pol_, Feb 09 2017
%e A279286 The first record of height difference is between the levels 1 and 2 of the pyramid (starting from the top), at the point (1,1) of the main diagonal of the top view of the pyramid, so a(1) = 1.
%e A279286 The second record of height difference is between the levels 2 and 4, at the point (2,2) of the main diagonal of the top view of the pyramid, so a(2) = 2.
%e A279286 The third record of height difference is between the levels 9 and 12, at the point (7,7) of the main diagonal of the top view of the pyramid, so a(3) = 7.
%e A279286 The fourth record of height difference is between the levels 20 and 24, at the point (15,15) of the main diagonal of the top view of the pyramid, so a(4) = 15.
%e A279286 Illustration of the diagram of the symmetries of sigma (n = 1..16), which is also the top view of the pyramid described in A245092, and it is also a quadrant of the top view of the pyramid described in A244050:
%e A279286 .     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A279286 .    |_| | | | | | | | | | | | | | | |
%e A279286 .    |_ _|_| | | | | | | | | | | | | |
%e A279286 .    |_ _|  _|_| | | | | | | | | | | |
%e A279286 .    |_ _ _|    _|_| | | | | | | | | |
%e A279286 .    |_ _ _|  _|  _ _|_| | | | | | | |
%e A279286 .    |_ _ _ _|  _| |  _ _|_| | | | | |
%e A279286 .    |_ _ _ _| |_ _|_|    _ _|_| | | |
%e A279286 .    |_ _ _ _ _|  _|     |  _ _ _|_| |
%e A279286 .    |_ _ _ _ _| |      _|_| |  _ _ _|
%e A279286 .    |_ _ _ _ _ _|  _ _|    _| |
%e A279286 .    |_ _ _ _ _ _| |  _|  _|  _|
%e A279286 .    |_ _ _ _ _ _ _| |_ _|  _|
%e A279286 .    |_ _ _ _ _ _ _| |  _ _|
%e A279286 .    |_ _ _ _ _ _ _ _| |
%e A279286 .    |_ _ _ _ _ _ _ _| |
%e A279286 .    |_ _ _ _ _ _ _ _ _|
%e A279286 ...
%t A279286 a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
%t A279286 a279286[b_] := Module[{centers={{1, 1}}, acc={1}, k=2, cPrev=1, cCur, len}, While[k<=b, cCur=a240542[k]; If[Last[acc]==cCur, AppendTo[acc, cCur], len=Length[acc]; If[First[Last[centers]]<len, AppendTo[centers, {len, cPrev}]]; acc={cCur}; cPrev=cCur]; k++]; Last[Transpose[centers]]]
%t A279286 a279286[5000000] (* data *)
%t A279286 (* _Hartmut F. W. Hoft_, Feb 08 2017 *)
%Y A279286 Where records occur in A259179, (was the original Name).
%Y A279286 Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A240542, A245092, A244050, A262626, A282197.
%K A279286 nonn,hard,more
%O A279286 1,2
%A A279286 _Omar E. Pol_, Dec 09 2016
%E A279286 a(7)-a(28) from _Hartmut F. W. Hoft_, Feb 08 2017
%E A279286 New Name from _Omar E. Pol_, Feb 09 2017