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.

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, 26, 25, 29, 36, 30, 35, 31, 34, 32, 33, 37, 45, 38, 44, 39, 43, 40, 42, 41, 46, 55, 47, 54, 48, 53, 49, 52, 50, 51, 56, 66, 57, 65, 58, 64, 59, 63, 60, 62, 61, 67, 78, 68, 77, 69, 76

Offset: 1

Author

Clark Kimberling, Jul 28 2000

Keywords

• nonn
• tabl

Comments

Arrange the quotients F(i)/F(j) of Fibonacci numbers, for 2<=i

Examples

			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)).
Triangle starts:
  1;
  2, 3;
  4, 6, 5;
  7,10, 8, 9;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Reflection of the array in A057028 about its central column, a permutation of the natural numbers.

Inverse permutation to A064578. Central column: A057029.

Column 1 is A000124, column 2 is A000217.

Row sums are A006003.

Programs

Mathematica

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 *)

Formula

From Werner Schulte, Sep 09 2024: (Start)

T(n, k) = (n^2 + (-1)^k * (n - k) + (3 + (-1)^k) / 2) / 2.

T(n, 1) = (n^2 - n + 2) / 2 = A000124(n).

T(n, 2) = (n^2 + n) / 2 = A000217(n) for n >= 2.

T(n, k) = T(n, k-2) - (-1)^k for 3 <= k <= n. (End)

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

Extensions

Corrected and extended by Vladeta Jovovic, Oct 18 2001

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