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