A271355 Triangular array: T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 7, 4, 3, 2, 1, 11, 7, 4, 3, 2, 1, 18, 11, 7, 4, 3, 2, 1, 29, 18, 11, 7, 4, 3, 2, 1, 47, 29, 18, 11, 7, 4, 3, 2, 1, 76, 47, 29, 18, 11, 7, 4, 3, 2, 1, 123, 76, 47, 29, 18, 11, 7, 4, 3, 2, 1, 199, 123, 76, 47, 29, 18, 11, 7, 4, 3

Offset: 1

Author

Clark Kimberling, May 01 2016

Keywords

• nonn
• easy
• tabl

Comments

Row n consists of the first n numbers of A169985 = (1,2,3,4,7,... ) in reverse order; these are the Lucas numbers, A000032, with order of initial two terms reversed. Every column of the triangle is A169985.

Examples

			First six rows:
  1
  2   1
  3   2   1
  4   3   2   1
  7   4   3   2   1
  11  7   4   3   2   1
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A169985, A000032, A000045, A104762

Programs

Mathematica

r = N[(1 + Sqrt[5])/2, 100]; s = N[(1 - Sqrt[5])/2, 100];
t = Table[Abs[Round[(r^n)*(s^k)]], {n, 0, 15}, {k, 1, n}];
Flatten[t]  (* A271355, sequence *)
TableForm[t]  (* A271355, array *)

Formula

T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

T(k+j-1,j) = A000032(k) = k-th Lucas number, for k >= 2.