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A259177 Triangle read by rows T(n,k) in which row n lists the even-indexed terms of n-th row of triangle A237593.

%I A259177 #47 Apr 27 2021 09:41:09
%S A259177 1,2,1,2,1,3,2,3,1,1,4,2,1,4,2,1,5,2,2,5,2,1,1,6,3,1,1,6,2,1,2,7,3,1,
%T A259177 2,7,3,2,1,8,3,1,1,2,8,3,1,1,2,9,4,1,1,2,9,3,2,1,2,10,4,2,1,2,10,4,1,
%U A259177 2,2,11,4,1,1,1,3,11,4,2,1,1,2,12,5,2,1,1,2,12,4,2,1,1,3,13,5,1,1,2,3,13,5,2,1,2,2,14
%N A259177 Triangle read by rows T(n,k) in which row n lists the even-indexed terms of n-th row of triangle A237593.
%C A259177 Row n has length A003056(n) hence column k starts in row A000217(k).
%C A259177 Row n is a permutation of the n-th row of A237591 for some n, hence the sequence is a permutation of A237591.
%e A259177 Written as an irregular triangle the sequence begins:
%e A259177 1;
%e A259177 2;
%e A259177 1, 2;
%e A259177 1, 3;
%e A259177 2, 3;
%e A259177 1, 1, 4;
%e A259177 2, 1, 4;
%e A259177 2, 1, 5;
%e A259177 2, 2, 5;
%e A259177 2, 1, 1, 6;
%e A259177 3, 1, 1, 6;
%e A259177 2, 1, 2, 7;
%e A259177 3, 1, 2, 7;
%e A259177 3, 2, 1, 8;
%e A259177 3, 1, 1, 2, 8;
%e A259177 3, 1, 1, 2, 9;
%e A259177 ...
%e A259177 Illustration of initial terms (side view of the pyramid):
%e A259177 Row                                 _
%e A259177 1                                 _|_|
%e A259177 2                               _|_ _|
%e A259177 3                             _|_|_ _|
%e A259177 4                           _|_|_ _ _|
%e A259177 5                         _|_ _|_ _ _|
%e A259177 6                       _|_|_|_ _ _ _|
%e A259177 7                     _|_ _|_|_ _ _ _|
%e A259177 8                   _|_ _|_|_ _ _ _ _|
%e A259177 9                 _|_ _|_ _|_ _ _ _ _|
%e A259177 10              _|_ _|_|_|_ _ _ _ _ _|
%e A259177 11            _|_ _ _|_|_|_ _ _ _ _ _|
%e A259177 12          _|_ _|_|_ _|_ _ _ _ _ _ _|
%e A259177 13        _|_ _ _|_|_ _|_ _ _ _ _ _ _|
%e A259177 14      _|_ _ _|_ _|_|_ _ _ _ _ _ _ _|
%e A259177 15    _|_ _ _|_|_|_ _|_ _ _ _ _ _ _ _|
%e A259177 16   |_ _ _|_|_|_ _|_ _ _ _ _ _ _ _ _|
%e A259177 ...
%e A259177 The above structure represents the first 16 levels (starting from the top) of one of the side views of the infinite stepped pyramid described in A245092. For another side view see A259176.
%e A259177 .
%e A259177 Illustration of initial terms (partial front view of the pyramid):
%e A259177 Row                                 _
%e A259177 1                                  |_|_
%e A259177 2                                 _|_ _|_
%e A259177 3                                |_| |_ _|_
%e A259177 4                               _|_| |_ _ _|_
%e A259177 5                              |_ _|_  |_ _ _|_
%e A259177 6                             _|_| |_| |_ _ _ _|_
%e A259177 7                            |_ _| |_|   |_ _ _ _|_
%e A259177 8                           _|_ _| |_|_  |_ _ _ _ _|_
%e A259177 9                          |_ _|  _|_ _|   |_ _ _ _ _|_
%e A259177 10                        _|_ _| |_| |_|   |_ _ _ _ _ _|_
%e A259177 11                       |_ _ _| |_| |_|_    |_ _ _ _ _ _|_
%e A259177 12                      _|_ _|   |_| |_ _|   |_ _ _ _ _ _ _|_
%e A259177 13                     |_ _ _|  _|_| |_ _|     |_ _ _ _ _ _ _|_
%e A259177 14                    _|_ _ _| |_ _|_  |_|_    |_ _ _ _ _ _ _ _|_
%e A259177 15                   |_ _ _|   |_| |_| |_ _|     |_ _ _ _ _ _ _ _|_
%e A259177 16                   |_ _ _|   |_| |_| |_ _|     |_ _ _ _ _ _ _ _ _|
%e A259177 ...
%e A259177 A part of the hidden pattern of the symmetric representation of sigma emerges from the partial front view of the pyramid described in A245092.
%e A259177 For another partial front view see A259176. For the total front view see A237593.
%t A259177 (* function f[n,k] and its support functions are defined in A237593 *)
%t A259177 a259177[n_, k_] := f[n, 2*k]
%t A259177 TableForm[Table[a259177[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
%t A259177 Flatten[Table[a259177[n, k], {n, 1, 26}, {k, 1, [n]}]] (* sequence data *)
%t A259177 (* _Hartmut F. W. Hoft_, Mar 06 2017 *)
%o A259177 (PARI) row(n) = (sqrt(8*n + 1) - 1)\2;
%o A259177 s(n, k) = ceil((n + 1)/k - (k + 1)/2) - ceil((n + 1)/(k + 1) - (k + 2)/2);
%o A259177 T(n, k) = if(k<=row(n), s(n, k), s(n, 2*row(n) + 1 - k));
%o A259177 a259177(n, k) = T(n, 2*k);
%o A259177 for(n=1, 26, for(k=1, row(n), print1(a259177(n, k),", ");); print();) \\ _Indranil Ghosh_, Apr 21 2017
%o A259177 (Python)
%o A259177 from sympy import sqrt
%o A259177 import math
%o A259177 def row(n): return int(math.floor((sqrt(8*n + 1) - 1)/2))
%o A259177 def s(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2)) - int(math.ceil((n + 1)/(k + 1) - (k + 2)/2))
%o A259177 def T(n, k): return s(n, k) if k<=row(n) else s(n, 2*row(n) + 1 - k)
%o A259177 def a259177(n, k): return T(n, 2*k)
%o A259177 for n in range(1, 27): print([a259177(n, k) for k in range(1, row(n) + 1)]) # _Indranil Ghosh_, Apr 21 2017
%Y A259177 Bisection of A237593.
%Y A259177 Row sums give A000027.
%Y A259177 Mirror of A259176 which is another bisection of A237593.
%Y A259177 Cf. A000203, A000217, A003056, A024916, A175254, A196020, A236104, A237270, A237271, A237591, A244580, A245092, A249351, A259179, A261350.
%K A259177 nonn,tabf
%O A259177 1,2
%A A259177 _Omar E. Pol_, Aug 15 2015
%E A259177 Better definition from _Omar E. Pol_, Apr 26 2021