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A259176 Triangle read by rows T(n,k) in which row n lists the odd-indexed terms of n-th row of triangle A237593.

%I A259176 #57 Apr 27 2021 09:40:39
%S A259176 1,2,2,1,3,1,3,2,4,1,1,4,1,2,5,1,2,5,2,2,6,1,1,2,6,1,1,3,7,2,1,2,7,2,
%T A259176 1,3,8,1,2,3,8,2,1,1,3,9,2,1,1,3,9,2,1,1,4,10,2,1,2,3,10,2,1,2,4,11,2,
%U A259176 2,1,4,11,3,1,1,1,4,12,2,1,1,2,4,12,2,1,1,2,5,13,3,1,1,2,4,13,3,2,1,1,5,14,2,2,1,2,5
%N A259176 Triangle read by rows T(n,k) in which row n lists the odd-indexed terms of n-th row of triangle A237593.
%C A259176 Row n has length A003056(n) hence column k starts in row A000217(k).
%C A259176 Row n is a permutation of the n-th row of A237591 for some n, hence the sequence is a permutation of A237591.
%e A259176 Written as an irregular triangle the sequence begins:
%e A259176 1;
%e A259176 2;
%e A259176 2, 1;
%e A259176 3, 1;
%e A259176 3, 2;
%e A259176 4, 1, 1;
%e A259176 4, 1, 2;
%e A259176 5, 1, 2;
%e A259176 5, 2, 2;
%e A259176 6, 1, 1, 2;
%e A259176 6, 1, 1, 3;
%e A259176 7, 2, 1, 2;
%e A259176 7, 2, 1, 3;
%e A259176 8, 1, 2, 3;
%e A259176 8, 2, 1, 1, 3;
%e A259176 9, 2, 1, 1, 3;
%e A259176 ...
%e A259176 Illustration of initial terms (side view of the pyramid):
%e A259176 Row   _
%e A259176 1    |_|_
%e A259176 2    |_ _|_
%e A259176 3    |_ _|_|_
%e A259176 4    |_ _ _|_|_
%e A259176 5    |_ _ _|_ _|_
%e A259176 6    |_ _ _ _|_|_|_
%e A259176 7    |_ _ _ _|_|_ _|_
%e A259176 8    |_ _ _ _ _|_|_ _|_
%e A259176 9    |_ _ _ _ _|_ _|_ _|_
%e A259176 10   |_ _ _ _ _ _|_|_|_ _|_
%e A259176 11   |_ _ _ _ _ _|_|_|_ _ _|_
%e A259176 12   |_ _ _ _ _ _ _|_ _|_|_ _|_
%e A259176 13   |_ _ _ _ _ _ _|_ _|_|_ _ _|_
%e A259176 14   |_ _ _ _ _ _ _ _|_|_ _|_ _ _|_
%e A259176 15   |_ _ _ _ _ _ _ _|_ _|_|_|_ _ _|_
%e A259176 16   |_ _ _ _ _ _ _ _ _|_ _|_|_|_ _ _|
%e A259176 ...
%e A259176 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 A259177.
%e A259176 .
%e A259176 Illustration of initial terms (partial front view of the pyramid):
%e A259176 Row                                 _
%e A259176 1                                 _|_|
%e A259176 2                               _|_ _|_
%e A259176 3                             _|_ _| |_|
%e A259176 4                           _|_ _ _| |_|_
%e A259176 5                         _|_ _ _|  _|_ _|
%e A259176 6                       _|_ _ _ _| |_| |_|_
%e A259176 7                     _|_ _ _ _|   |_| |_ _|
%e A259176 8                   _|_ _ _ _ _|  _|_| |_ _|_
%e A259176 9                 _|_ _ _ _ _|   |_ _|_  |_ _|
%e A259176 10              _|_ _ _ _ _ _|   |_| |_| |_ _|_
%e A259176 11            _|_ _ _ _ _ _|    _|_| |_| |_ _ _|
%e A259176 12          _|_ _ _ _ _ _ _|   |_ _| |_|   |_ _|_
%e A259176 13        _|_ _ _ _ _ _ _|     |_ _| |_|_  |_ _ _|
%e A259176 14      _|_ _ _ _ _ _ _ _|    _|_|  _|_ _| |_ _ _|_
%e A259176 15    _|_ _ _ _ _ _ _ _|     |_ _| |_| |_|   |_ _ _|
%e A259176 16   |_ _ _ _ _ _ _ _ _|     |_ _| |_| |_|   |_ _ _|
%e A259176 ...
%e A259176 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 A259176 For another partial front view see A259177. For the total front view see A237593.
%t A259176 (* function f[n,k] and its support functions are defined in A237593 *)
%t A259176 a259176[n_, k_] := f[n, 2*k-1]
%t A259176 TableForm[Table[a259176[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
%t A259176 Flatten[Table[a259176[n, k], {n, 1, 26}, {k, 1, [n]}]] (* sequence data *)
%t A259176 (* _Hartmut F. W. Hoft_, Mar 06 2017 *)
%o A259176 (PARI) row(n) = (sqrt(8*n + 1) - 1)\2;
%o A259176 s(n, k) = ceil((n + 1)/k - (k + 1)/2) - ceil((n + 1)/(k + 1) - (k + 2)/2);
%o A259176 T(n, k) = if(k<=row(n), s(n, k), s(n, 2*row(n) + 1 - k));
%o A259176 a259177(n, k) = T(n, 2*k - 1);
%o A259176 for(n=1, 26, for(k=1, row(n), print1(a259177(n, k),", ");); print();)  \\ _Indranil Ghosh_, Apr 21 2017
%o A259176 (Python)
%o A259176 from sympy import sqrt
%o A259176 import math
%o A259176 def row(n): return int(math.floor((sqrt(8*n + 1) - 1)/2))
%o A259176 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 A259176 def T(n, k): return s(n, k) if k<=row(n) else s(n, 2*row(n) + 1 - k)
%o A259176 def a259177(n, k): return T(n, 2*k - 1)
%o A259176 for n in range(1, 11): print([a259177(n, k) for k in range(1, row(n) + 1)]) # _Indranil Ghosh_, Apr 21 2017
%Y A259176 Bisection of A237593.
%Y A259176 Row sums give A000027.
%Y A259176 For the mirror see A259177 which is another bisection of A237593.
%Y A259176 Cf. A000203, A000217, A003056, A024916, A175254, A196020, A236104, A237270, A237271, A237591, A244580, A245092, A249351, A259179, A261350.
%K A259176 nonn,tabf
%O A259176 1,2
%A A259176 _Omar E. Pol_, Aug 15 2015
%E A259176 Better definition from _Omar E. Pol_, Apr 26 2021