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A057027 Triangle T read by rows: row n consists of the numbers C(n,2)+1 to C(n+1,2); numbers in odd-numbered places form an increasing sequence and the others a decreasing sequence.

%I A057027 #36 Sep 11 2024 00:43:27
%S A057027 1,2,3,4,6,5,7,10,8,9,11,15,12,14,13,16,21,17,20,18,19,22,28,23,27,24,
%T A057027 26,25,29,36,30,35,31,34,32,33,37,45,38,44,39,43,40,42,41,46,55,47,54,
%U A057027 48,53,49,52,50,51,56,66,57,65,58,64,59,63,60,62,61,67,78,68,77,69,76
%N A057027 Triangle T read by rows: row n consists of the numbers C(n,2)+1 to C(n+1,2); numbers in odd-numbered places form an increasing sequence and the others a decreasing sequence.
%C A057027 Arrange the quotients F(i)/F(j) of Fibonacci numbers, for 2<=i<j<=n, in increasing order. Then the positions of the F(i)/F(n-k) are the first n-k-2 terms of the diagonal T(i,i-k), for k=0,1,2,...,n-3.
%H A057027 Michael De Vlieger, <a href="/A057027/b057027.txt">Table of n, a(n) for n = 1..10000</a>
%H A057027 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A057027 From _Werner Schulte_, Sep 09 2024: (Start)
%F A057027 T(n, k) = (n^2 + (-1)^k * (n - k) + (3 + (-1)^k) / 2) / 2.
%F A057027 T(n, 1) = (n^2 - n + 2) / 2 = A000124(n).
%F A057027 T(n, 2) = (n^2 + n) / 2 = A000217(n) for n >= 2.
%F A057027 T(n, k) = T(n, k-2) - (-1)^k  for 3 <= k <= n. (End)
%F A057027 G.f.: x*y*(1 + x*(y - 1) - x^4*(y - 1)*y^2 + x^5*y^3 + x^3*y*(y^2 - y - 1) - x^2*(y^2 + y - 1))/((1 - x)^3*(1 - x*y)^3*(1 + x*y)). - _Stefano Spezia_, Sep 10 2024
%e A057027 For n=6, the ordered quotients are 1/8, 1/5, 2/8, 1/3, 3/8, 2/5, 1/2, 3/5, 5/8, 2/3; the positions of 1/5, 2/5, 3/5 are 2, 6, 8 (first terms of diagonal T(i, i-1)).
%e A057027 Triangle starts:
%e A057027   1;
%e A057027   2, 3;
%e A057027   4, 6, 5;
%e A057027   7,10, 8, 9;
%e A057027   ...
%t A057027 nn= 12; t = Table[Range[Binomial[n, 2] + 1, Binomial[n + 1, 2]], {n, nn}]; Table[t[[n, If[OddQ@ k, Ceiling[k/2], -k/2] ]], {n, nn}, {k, n}] // Flatten (* _Michael De Vlieger_, Jul 02 2016 *)
%Y A057027 Reflection of the array in A057028 about its central column, a permutation of the natural numbers.
%Y A057027 Inverse permutation to A064578. Central column: A057029.
%Y A057027 Column 1 is A000124, column 2 is A000217.
%Y A057027 Row sums are A006003.
%K A057027 nonn,tabl
%O A057027 1,2
%A A057027 _Clark Kimberling_, Jul 28 2000
%E A057027 Corrected and extended by _Vladeta Jovovic_, Oct 18 2001