A279286 - OEIS

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 #69 Feb 09 2017 10:25:21

%S 1,2,7,15,52,102,296,371,455,929,1853,2034,4517,4797,5829,6146,6948,

%T 17577,18915,60349,78369,85171,123788,128596,415355,906771,1308771,

%U 3329668

%N 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 Is this sequence infinite?

%C First differs from A282197 (another version) at a(19). - _Omar E. Pol_, Feb 08 2017

%C 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 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 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 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 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 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 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

%e . |_| | | | | | | | | | | | | | | |

%e . |_ _|_| | | | | | | | | | | | | |

%e . |_ _| _|_| | | | | | | | | | | |

%e . |_ _ _| _|_| | | | | | | | | |

%e . |_ _ _| _| _ _|_| | | | | | | |

%e . |_ _ _ _| _| | _ _|_| | | | | |

%e . |_ _ _ _| |_ _|_| _ _|_| | | |

%e . |_ _ _ _ _| _| | _ _ _|_| |

%e . |_ _ _ _ _| | _|_| | _ _ _|

%e . |_ _ _ _ _ _| _ _| _| |

%e . |_ _ _ _ _ _| | _| _| _|

%e . |_ _ _ _ _ _ _| |_ _| _|

%e . |_ _ _ _ _ _ _| | _ _|

%e . |_ _ _ _ _ _ _ _| |

%e . |_ _ _ _ _ _ _ _ _|

%e ...

%t a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]

%t 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[5000000] (* data *)

%t (* _Hartmut F. W. Hoft_, Feb 08 2017 *)

%Y Where records occur in A259179, (was the original Name).

%Y Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A240542, A245092, A244050, A262626, A282197.

%K nonn,hard,more

%O 1,2

%A _Omar E. Pol_, Dec 09 2016

%E a(7)-a(28) from _Hartmut F. W. Hoft_, Feb 08 2017

%E New Name from _Omar E. Pol_, Feb 09 2017