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 > = 0.

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

1,2

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.

Links

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

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 > = 0.

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

Example

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

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 *)

Crossrefs

Cf. A169985, A000032, A000045, A104762

Sequence in context: A200082 A052310 A052313 * A211230 A049085 A193173

Adjacent sequences: A271352 A271353 A271354 * A271356 A271357 A271358

Keyword

nonn,easy,tabl

Author

Clark Kimberling, May 01 2016

Status

approved